# 5.3 Determinants and Cramer s Rule

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Determinants and Cramer s Rule Unique Solution of a 2 2 System The 2 2 system (1) ax + by = e, cx + dy = f, has a unique solution provided = ad bc is nonzero, in which case the solution is given by (2) x = de bf ad bc, af ce y = ad bc. This result, called Cramer s Rule for 2 2 systems, is usually learned in college algebra as part of determinant theory. Determinants of Order 2 College algebra introduces matrix notation and determinant notation: ( ) a b a b A =, det(a) = c d c d. Evaluation of a 2 2 determinant is by Sarrus Rule: a b = ad bc. c d The boldface product ad is the product of the main diagonal entries and the other product bc is from the anti-diagonal. Cramer s 2 2 rule in determinant notation is e b a e f d c f (3) x = a b, y = a b. c d c d Unique Solution of an n n System Cramer s rule can be generalized to an n n system of equations A x = b or a 11 x 1 + a 12 x a 1n x n = b 1, a 21 x 1 + a 22 x a 2n x n = b 2, (4).... a n1 x 1 + a n2 x a nn x n = b n.

2 5.3 Determinants and Cramer s Rule 291 System (4) has a unique solution provided the determinant of coefficients = det(a) is nonzero, in which case the solution is given by (5) x 1 = 1, x 2 = 2,..., x n = n. The determinant j equals det(b j ) where matrix B j is matrix A with column j replaced by b = (b 1,..., b n ), which is the right side of system (4). The result is called Cramer s Rule for n n systems. Determinants will be defined shortly; intuition from the 2 2 case and Sarrus rule should suffice for the moment. Determinant Notation for Cramer s Rule. of coefficients for system A x = b is denoted by a 11 a 12 a 1n a 21 a 22 a 2n (6) =.... a n1 a n2 a nn The determinant The other n determinants in Cramer s rule (5) are given by (7) 1 = b 1 a 12 a 1n b 2 a 22 a 2n... b n a n2 a nn,..., n = a 11 a 12 b 1 a 21 a 22 b 2... a n1 a n2 b n The literature is filled with conflicting notations for matrices, vectors and determinants. The reader should take care to use vertical bars only for determinants and absolute values, e.g., A makes sense for a matrix A or a constant A. For clarity, the notation det(a) is preferred, when A is a matrix. The notation A implies that a determinant is a number, computed by A = ±A when n = 1, and A = a 11 a 22 a 12 a 21 when n = 2. For n 3, A is computed by similar but increasingly complicated formulas; see Sarrus rule and the four properties below.. Sarrus Rule for 3 3 Matrices. College algebra supplies the following formula for the determinant of a 3 3 matrix A: a 11 a 12 a 13 det(a) = a 21 a 22 a 23 (8) a 31 a 32 a 33 = a 11 a 22 a 33 + a 21 a 32 a 13 + a 31 a 12 a 23 a 11 a 32 a 23 a 21 a 12 a 33 a 31 a 22 a 13.

3 292 The number det(a) can be computed by an algorithm similar to the one for 2 2 matrices, as in Figure 8. We remark that no further generalizations are possible: there is no Sarrus rule for 4 4 or larger matrices! d a 11 a 12 a 13 e a 21 a 31 a 32 a 33 a a 11 a 22 a 12 a 23 a 13 b a 21 a 22 a 23 c f Figure 8. Sarrus rule for 3 3 matrices, which gives det(a) = (a + b + c) (d + e + f). College Algebra Definition of Determinant. definition is the formula (9) det(a) = σ S n ( 1) parity(σ) a 1σ1 a nσn. The impractical In the formula, a ij denotes the element in row i and column j of the matrix A. The symbol σ = (σ 1,..., σ n ) stands for a rearrangement of the subscripts 1, 2,..., n and S n is the set of all possible rearrangements. The nonnegative integer parity(σ) is determined by counting the minimum number of pairwise interchanges required to assemble the list of integers σ 1,..., σ n into natural order 1,..., n. A consequence of (9) is the relation det(a) = det(a T ) where A T means the transpose of A, obtained by swapping rows and columns. This relation implies that all determinant theory results for rows also apply to columns. Formula (9) reproduces the definition for 3 3 matrices given in equation (8). We will have no computational use for (9). For computing the value of a determinant, see below four properties and cofactor expansion. Four Properties. The definition of determinant (9) implies the following four properties: Triangular Swap Combination Multiply The value of det(a) for either an upper triangular or a lower triangular matrix A is the product of the diagonal elements: det(a) = a 11 a 22 a nn. If B results from A by swapping two rows, then det(a) = ( 1) det(b). The value of det(a) is unchanged by adding a multiple of a row to a different row. If one row of A is multiplied by constant c to create matrix B, then det(b) = c det(a).

4 5.3 Determinants and Cramer s Rule 293 It is known that these four rules suffice to compute the value of any n n determinant. The proof of the four properties is delayed until page 301. Elementary Matrices and the Four Rules. stated in terms of elementary matrices as follows. The rules can be Triangular Swap Combination Multiply The value of det(a) for either an upper triangular or a lower triangular matrix A is the product of the diagonal elements: det(a) = a 11 a 22 a nn. This is a one-arrow Sarrus rule valid for dimension n. If E is an elementary matrix for a swap rule, then det(ea) = ( 1) det(a). If E is an elementary matrix for a combination rule, then det(ea) = det(a). If E is an elementary matrix for a multiply rule with multiplier c 0, then det(ea) = c det(a). Since det(e) = 1 for a combination rule, det(e) = 1 for a swap rule and det(e) = c for a multiply rule with multiplier c 0, it follows that for any elementary matrix E there is the determinant multiplication rule det(ea) = det(e) det(a). Additional Determinant Rules. The following rules make for efficient evaluation of certain special determinants. The results are stated for rows, but they also hold for columns, because det(a) = det(a T ). Zero row If one row of A is zero, then det(a) = 0. Duplicate rows If two rows of A are identical, then det(a) = 0. RREF I If rref(a) I, then det(a) = 0. Common factor Row linearity The relation det(a) = c det(b) holds, provided A and B differ only in one row, say row j, for which row(a, j) = c row(b, j). The relation det(a) = det(b) + det(c) holds, provided A, B and C differ only in one row, say row j, for which row(a, j) = row(b, j) + row(c, j). The proofs of these properties are delayed until page 301. Cofactor Expansion The special subject of cofactor expansions is used to justify Cramer s rule and to provide an alternative method for computation of determinants. There is no claim that cofactor expansion is efficient, only that it is possible, and different than Sarrus rule or the use of the four properties.

5 294 Background from College Algebra. The cofactor expansion theory is most easily understood from the college algebra topic, where the dimension is 3 and row expansion means the following formulas are valid: a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 32 a 33 a = a 11 (+1) 22 a 23 a 32 a 33 + a a 21 a 23 12( 1) a 31 a 33 + a a 21 a 22 13(+1) a 31 a 32 a = a 21 ( 1) 12 a 13 a 32 a 33 + a a 11 a 13 22(+1) a 31 a 33 + a a 11 a 12 23( 1) a 31 a 32 a = a 31 (+1) 12 a 13 a 22 a 23 + a a 11 a 13 32( 1) a 21 a 23 + a a 11 a 12 33(+1) a 21 a 22 The formulas expand a 3 3 determinant in terms of 2 2 determinants, along a row of A. The attached signs ±1 are called the checkerboard signs, to be defined shortly. The 2 2 determinants are called minors of the 3 3 determinant A. The checkerboard sign together with a minor is called a cofactor. These formulas are generally used when a row has one or two zeros, making it unnecessary to evaluate one or two of the 2 2 determinants in the expansion. To illustrate, row 1 expansion gives = 3(+1) = 60. A clever time saving choice is always a row which has the most zeros, although success does not depend upon cleverness. What has been said for rows also applies to columns, due to the transpose formula A = A T. Minors and Cofactors. The (n 1) (n 1) determinant obtained from det(a) by striking out row i and column j is called the (i, j) minor of A and denoted minor(a, i, j) (M ij is common in literature). The (i, j) cofactor of A is cof(a, i, j) = ( 1) i+j minor(a, i, j). Multiplicative factor ( 1) i+j is called the checkerboard sign, because its value can be determined by counting plus, minus, plus, etc., from location (1, 1) to location (i, j) in any checkerboard fashion. Expansion of Determinants by Cofactors. The formulas are (10) det(a) = a kj cof(a, k, j), det(a) = a il cof(a, i, l), j=1 i=1

6 5.3 Determinants and Cramer s Rule 295 where 1 k n, 1 l n. The first expansion in (10) is called a cofactor row expansion and the second is called a cofactor column expansion. The value cof(a, i, j) is the cofactor of element a ij in det(a), that is, the checkerboard sign times the minor of a ij. The proof of expansion (10) is delayed until page 301. The Adjugate Matrix. The adjugate adj(a) of an n n matrix A is the transpose of the matrix of cofactors, adj(a) = cof(a, 1, 1) cof(a, 1, 2) cof(a, 1, n) cof(a, 2, 1) cof(a, 2, 2) cof(a, 2, n)... cof(a, n, 1) cof(a, n, 2) cof(a, n, n) T A cofactor cof(a, i, j) is the checkerboard sign ( 1) i+j times the corresponding minor determinant minor(a, i, j). In the 2 2 case,. adj ( a11 a 12 a 21 a 22 ) ( a22 a = 12 a 21 a 11 ) In words: swap the diagonal elements and change the sign of the off diagonal elements. The Inverse Matrix. The adjugate appears in the formula for the inverse matrix A 1 : ( ) 1 ( ) a11 a 12 1 a22 a = 12. a 21 a 22 a 11 a 22 a 12 a 21 a 21 a 11 This formula is verified by direct matrix multiplication: ( ) ( ) ( ) a11 a 12 a22 a = (a a 21 a 22 a 21 a 11 a 22 a 12 a 21 ) For an n n matrix, A adj(a) = det(a) I, which gives the formula A 1 = 1 det(a) cof(a, 1, 1) cof(a, 1, 2) cof(a, 1, n) cof(a, 2, 1) cof(a, 2, 2) cof(a, 2, n)... cof(a, n, 1) cof(a, n, 2) cof(a, n, n) The proof of A adj(a) = det(a) I is delayed to page 303. T Elementary Matrices. An elementary matrix E is the result of applying a combination, multiply or swap rule to the identity matrix. This definition implies that an elementary matrix is the identity matrix with a minor change applied, to wit:

7 296 Combination Change an off-diagonal zero of I to c. Multiply Change a diagonal one of I to multiplier m 0. Swap Swap two rows of I. Theorem 9 (Determinants and Elementary Matrices) Let E be an n n elementary matrix. Then Combination det(e) = 1 Multiply det(e) = m for multiplier m. Swap det(e) = 1 Product det(ex) = det(e) det(x) for all n n matrices X. Theorem 10 (Determinants and Invertible Matrices) Let A be a given invertible matrix. Then ( 1) s det(a) = m 1 m 2 m r where s is the number of swap rules applied and m 1, m 2,..., m r are the nonzero multipliers used in multiply rules when A is reduced to rref(a). Determinant Product Rule. The determinant rules of combination, multiply and swap imply that det(ex) = det(e) det(x) for elementary matrices E and square matrices X. We show that a more general relationship holds. Theorem 11 (Determinant Product Rule) Let A and B be given n n matrices. Then Proof: det(ab) = det(a) det(b). Used in the proof is the equivalence of invertibility of a square matrix C with det(c) 0 and rref(c) = I. Assume one of A or B has zero determinant. Then det(a) det(b) = 0. If det(b) = 0, then Bx = 0 has infinitely many solutions, in particular a nonzero solution x. Multiply Bx = 0 by A, then ABx = 0 which implies AB is not invertible. Then the identity det(ab) = det(a) det(b) holds, because both sides are zero. If det(b) 0 but det(a) = 0, then there is a nonzero y with Ay = 0. Define x = AB 1 y. Then ABx = Ay = 0, with x 0, which implies the identity holds.. This completes the proof when one of A or B is not invertible. Assume A, B are invertible and then C = AB is invertible. In particular, rref(a 1 ) = rref(b 1 ) = I. Write I = rref(a 1 ) = E 1 E 2 E k A 1 and I = rref(b 1 ) = F 1 F 2 F m B 1 for elementary matrices E i, F j. Then (11) AB = E 1 E 2 E k F 1 F 2 F m.

9 298 2 Example (Four Properties) Apply the four properties of a determinant to justify the formula det = Solution: Let D denote the value of the determinant. Then D = det Given = det Combination rule: row 1 subtracted from the others. = 6 det = 6 det = 6 det = 6 det = 6 det Multiply rule. Combination rule: add row 1 to row 3, then add twice row 2 to row 1. Swap rule. Multiply rule. Combination rule. = 6(1)( 1)( 4) Triangular rule. = 24 Formula verified. 3 Example (Hybrid Method) Justify by cofactor expansion and the four properties the identity det 11 5 a = 5(6a b) b Solution: Let D denote the value of the determinant. Then D = det 11 5 a Given b = det 1 0 a Combination: subtract row 1 from the 0 3 b other rows.

10 5.3 Determinants and Cramer s Rule a = det 1 0 a 0 3 b ( ) 5 10a = (1)( 1) det 3 b Combination: add 10 times row 2 to row 1. Cofactor expansion on column 1. = (1)( 1)(5b 30a) Sarrus rule for n = 2. = 5(6a b). Formula verified. 4 Example (Cramer s Rule) Solve by Cramer s rule the system of equations 2x 1 + 3x 2 + x 3 x 4 = 1, x 1 + x 2 x 4 = 1, 3x 2 + x 3 + x 4 = 3, x 1 + x 3 x 4 = 0, verifying x 1 = 1, x 2 = 0, x 3 = 1, x 4 = 2. Solution: Form the four determinants 1,..., 4 from the base determinant as follows: = det , = det 3 = det ,, 2 = det 4 = det Five repetitions of the methods used in the previous examples give the answers = 2, 1 = 2, 2 = 0, 3 = 2, 4 = 4, therefore Cramer s rule implies the solution x i = i /, 1 i 4. Then x 1 = 1, x 2 = 0, x 3 = 1, x 4 = 2. Maple code. The details of the computation above can be checked in computer algebra system maple as follows. with(linalg): A:=matrix([ [2, 3, 1, -1], [1, 1, 0, -1], [0, 3, 1, 1], [1, 0, 1, -1]]); Delta:= det(a); B1:=matrix([ [ 1, 3, 1, -1], [-1, 1, 0, -1], [ 3, 3, 1, 1], [ 0, 0, 1, -1]]); Delta1:=det(B1); x[1]:=delta1/delta;,.

11 300 An Applied Definition of Determinant To be developed here is another way to look at formula (9), which emphasizes the column and row structure of a determinant. The definition, which agrees with (9), leads to a short proof of the four properties, which are used to find the value of any determinant. Permutation Matrices. A matrix P obtained from the identity matrix I by swapping rows is called a permutation matrix. There are n! permutation matrices. To illustrate, the 3 3 permutation matrices are , 0 0 1, 1 0 0, 0 0 1, 1 0 0, Define for a permutation matrix P the determinant by det(p ) = ( 1) k where k is the least number of row swaps required to convert P to the identity. The number k satisfies r = k + 2m, where r is any count of row swaps that changes P to the identity, and m is some integer. Therefore, det(p ) = ( 1) k = ( 1) r. In the illustration, the corresponding determinants are 1, 1, 1, 1, 1, 1, as computed from det(p ) = ( 1) r, where r row swaps change P into I. It can be verified that det(p ) agrees with the value reported by formula (9). Each σ in (9) corresponds to a permutation matrix P with rows arranged in the order specified by σ. The summation in (9) for A = P has exactly one nonzero term. Sampled product. Let be given an n n matrix A and an n n permutation matrix P. The matrix P has ones (1) in exactly n locations. The sampled product A.P uses these locations to select entries from the matrix A, whereupon A.P is the product of those entries. More precisely, let A 1,..., An be the rows of A and let P 1,..., Pn be the rows of P. Define via the normal dot product ( ) the sampled product (15) A.P = (A 1 P 1 )(A 2 P 2 ) (A n P n ) = a 1σ1 a nσn, where the rows of P are rows σ 1,...,σ n of I. The formula implies that A.P is a linear function of the rows of A. A similar construction shows A.P is a linear function of the columns of A. Applied determinant formula. The determinant is defined by (16) det(a) = P det(p ) A.P,

12 5.3 Determinants and Cramer s Rule 301 where the summation extends over all possible permutation matrices P. The definition emphasizes the explicit linear dependence of the determinant upon the rows of A (or the columns of A). A tedious but otherwise routine justification shows that (9) and (16) give the same result. Verification of the Four Properties: Triangular. If A is n n triangular, then in (16) appears only one nonzero term, due to zero factors in the product A.P. The term that appears corresponds to P =identity, therefore A.P is the product of the diagonal elements of A. Since det(p ) = det(i) = 1, the result follows. A similar proof can be constructed from determinant definition (9). Swap. Let Q be obtained from I by swapping rows i and j. Let B = QA, so that B is A with rows i and j swapped. We must show det(a) = det(b). Observe that B.P = A.QP and det(qp ) = det(p ). The matrices QP over all possible P simply relist all permutation matrices, hence definition (16) implies the result. Combination. Let matrix B be obtained from matrix A by adding to row j the vector k times row i (i j). Then row(b, j) = row(a, j)+k row(a, i) and B.P = (B 1 P ) (B n P ) = A.P +k C.P, where C is the matrix obtained from A by replacing row(a, j) with row(a, i). Then C has equal rows row(c, i) = row(c, j) = row(a, i). By the swap rule applied to rows i and j, det(c) = det(c), or det(c) = 0. Add on P across B.P = A.P + k C.P to obtain det(b) = det(a) + k det(c). This implies det(b) = det(a). Multiply. Let matrices A and B have the same rows, except for some index i, row(b, i) = c row(a, i). Then B.P = c A.P. Add on P across this equation to obtain det(b) = c det(a). Verification of the Additional Rules: Duplicate rows. The swap rule applies to the two duplicate rows to give det(a) = det(a), hence det(a) = 0. Zero row. Apply the common factor rule with c = 2, possible since the row has all zero entries. Then det(a) = 2 det(a), giving det(a) = 0. Common factor and row linearity. The sampled product A.P is a linear function of each row, therefore the same is true of det(a). Derivations: Cofactors and Cramer s Rule Derivation of cofactor expansion (10): The column expansion formula is derived from the row expansion formula applied to the transpose. We consider only the derivation of the row expansion formula (10) for k = 1, because the case for general k is the same except for notation. The plan is to establish equality of the two sides of (10) for k = 1, which in terms of minor(a, 1, j) = ( 1) 1+j cof(a, 1, j) is the equality (17) det(a) = a 1j ( 1) 1+j minor(a, 1, j). j=1

13 302 The details require expansion of minor(a, 1, j) in (17) via the definition of determinant det(a) = σ ( 1)parity(σ) a 1σ1 a nσn. A typical term on the right in (17) after expansion looks like a 1j ( 1) 1+j ( 1) parity(α) a 2α2 a nαn. Here, α is a rearrangement of the set of n 1 elements consisting of 1,..., j 1, j + 1,..., n. Define σ = (j, α 2,..., α n ), which is a rearrangement of symbols 1,..., n. After parity(α) interchanges, α is changed into (1,..., j 1, j + 1,..., n) and therefore these same interchanges transform σ into (j, 1,..., j 1, j + 1,..., n). An additional j 1 interchanges will transform σ into natural order (1,..., n). This establishes, because of ( 1) j 1 = ( 1) j+1, the identity Collecting formulas gives ( 1) parity(σ) = ( 1) j 1+parity(α) = ( 1) j+1+parity(α). ( 1) parity(σ) a 1σ1 a nσn = a 1j ( 1) 1+j ( 1) parity(α) a 2α2 a nαn. Adding across this formula over all α and j gives a sum on the right which matches the right side of (17). Some additional thought reveals that the terms on the left add exactly to det(a), hence (17) is proved. Derivation of Cramer s Rule: The cofactor column expansion theory implies that the Cramer s rule solution of A x = b is given by (18) x j = j = 1 b k cof(a, k, j). k=1 We will verify that A x = b. Let E 1,..., E n be the rows of the identity matrix. The question reduces to showing that E p A x = b p. The details will use the fact (19) a pj cof(a, k, j) = j=1 { det(a) for k = p, 0 for k p, Equation (19) follows by cofactor row expansion, because the sum on the left is det(b) where B is matrix A with row k replaced by row p. If B has two equal rows, then det(b) = 0; otherwise, B = A and det(b) = det(a). E p A x = a pj x j j=1 = 1 = 1 j=1 k=1 a pj b k n k=1 b k cof(a, k, j) Apply formula (18). a pj cof(a, k, j) j=1 = b p Apply (19). Switch order of summation.

14 5.3 Determinants and Cramer s Rule 303 Derivation of A adj(a) = det(a)i: The proof uses formula (19). Consider column k of adj(a), denoted X, multiplied against matrix A, which gives n j=1 a 1j cof(a, k, j) AX n = j=1 a 2j cof(a, k, j).. n j=1 a nj cof(a, k, j) By formula (19), a ij cof(a, k, j) = j=1 { det(a) i = k, 0 i k. Therefore, A X is det(a) times column k of the identity I. This completes the proof. Three Properties that Define a Determinant Write the determinant det(a) in terms of the rows A 1,..., A n of the matrix A as follows: D 1 (A 1,..., A n ) = P det(p ) A.P. Already known is that D 1 (A 1,..., A n ) is a function D that satisfies the following three properties: Linearity D is linear in each argument A 1,..., A n. Swap D changes sign if two arguments are swapped. Equivalently, D = 0 if two arguments are equal. Identity D = 1 when A = I. The equivalence reported in swap is obtained by expansion, e.g., for n = 2, A 1 = A 2 implies D(A 1, A 2 ) = D(A 1, A 2 ) and hence D = 0. Similarly, D(A 1 +A 2, A 1 +A 2 ) = 0 implies by linearity that D(A 1, A 2 ) = D(A 2, A 1 ), which is the swap property for n = 2. It is less obvious that the three properties uniquely define the determinant, that is: Theorem 12 (Uniqueness) If D(A 1,..., A n ) satisfies the properties of linearity, swap and identity, then D(A 1,..., A n ) = det(a). Proof: The rows of the identity matrix I are denoted E 1,..., E n, so that for 1 j n we may write the expansion (20) A j = a j1 E 1 + a j2 E a jn E n. We illustrate the proof for the case n = 2:

15 304 D(A 1, A 2 ) = D(a 11 E 1 + a 12 E 2, A 2 ) By (20). = a 11 D(E 1, A 2 ) + a 12 D(E 2, A 2 ) By linearity. = a 11 a 22 D(E 1, E 2 ) + a 11 a 21 D(E 1, E 1 ) Repeat for A 2. + a 12 a 21 D(E 2, E 1 ) + a 12 a 22 D(E 2, E 2 ) The swap and identity properties give D(E 1, E 1 ) = D(E 2, E 2 ) = 0 and 1 = D(E 1, E 2 ) = D(E 2, E 1 ). Therefore, D(A 1, A 2 ) = a 11 a 22 a 12 a 21 and this implies that D(A 1, A 2 ) = det(a). The proof for general n depends upon the identity D(E σ1,..., E σn ) = ( 1) parity(σ) D(E 1,..., E n ) = ( 1) parity(σ) where σ = (σ 1,..., σ n ) is a rearrangement of the integers 1,..., n. This identity is implied by the swap and identity properties. Then, as in the case n = 2, linearity implies that D(A 1,..., A n ) = σ a 1σ 1 a nσn D(E σ1,..., E σn ) = σ ( 1)parity(σ) a 1σ1 a nσn = det(a). Exercises 5.3 Determinant Notation. Write formulae for x and y as quotients of 2 2 determinants. Do not evaluate the determinants! ( ) ( ) ( ) 1 1 x = 2 6 y 3 Unique Solution of a 2 2 System. Solve AX = b for X using Cramer s rule for 2 2 systems. 2. A = ( ), b = ( 1 1 Sarrus 2 2 rule. Evaluate det(a). ( ) A = 0 0 Sarrus rule 3 3. Evaluate det(a) A = Definition of Determinant. ) 5. Let A be 3 3 with first row all zeros. Use the college algebra definition of determinant to show that det(a) = 0. Four Properties. Evaluate det(a) using the four properties for determinants, page?? A = Elementary Matrices and the Four Rules. Find det(a). 7. A is 3 3 and obtained from the identity matrix I by three row swaps. 8. A is 7 7, obtained from I by swapping rows 1 and 2, then rows 4 and 1, then rows 1 and 3. More Determinant Rules. Cite the determinant rule that verifies det(a) = 0. Never expand det(a)!

16 5.3 Determinants and Cramer s Rule A = Cofactor Expansion and College Algebra. Evaluate det(a) using the most efficient cofactor expansion A = Minors and Cofactors. Write out and then evaluate the minor and cofactor of each element cited for the matrix A = x y 1 2 z 11. Row 1 and column 3. Cofactor Expansion. Use cofactors to evaluate det(a) A = Adjugate and Inverse Matrix. Find the adjugate of A and the inverse of A. Check the answers via the formula A adj(a) = det(a)i. ( 13. A = 14. A = ) Elementary Matrices. Find the determinant of the product A of elementary matrices. 15. Let A = E 1 E 2 be 9 9, where E 1 multiplies row 3 of the identity by 7 and E 2 swaps rows 3 and 5 of the identity. Determinants and Invertible Matrices. Test for invertibility of A. If invertible, then find A 1 by the best method: rref method or the adjugate formula. 16. A = Determinant Product Rule. Apply the product rule det(ab) = det(a) det(b). 17. Let det(a) = 5 and det(b) = 2. Find det(a 2 B 3 ). 18. Let det(a) = 4 and A(B 2A) = 0. Find det(b). 19. Let A = E 1 E 2 E 3 where E 1, E 2 are elementary swap matrices and E 3 is an elementary combination matrix. Find det(a). 20. Assume det(ab + A) = 0 and det(a) 0. Show that det(b + I) = 0. Cayley-Hamilton Theorem. 21. Let λ 2 2λ + 1 = 0 be the characteristic equation of a matrix A. Find a formula for A 2 in terms of A and I. 22. Let A be an n n triangular matrix with all diagonal entries zero. Prove that A n = 0. Applied Definition of Determinant. 23. Given A, find the sampled product for the permutation matrix P. A = 0 5 7, P = Determine the permutation matrices P required to evaluate det(a) when A is 4 4. Three Properties. 25. Assume n = 3. Prove that the three properties imply D = 0 when two rows are identical. 26. Assume n = 3. Prove that the three properties imply D = 0 when a row is zero.

### Determinants. Dr. Doreen De Leon Math 152, Fall 2015

Determinants Dr. Doreen De Leon Math 52, Fall 205 Determinant of a Matrix Elementary Matrices We will first discuss matrices that can be used to produce an elementary row operation on a given matrix A.

### 1 Determinants. Definition 1

Determinants The determinant of a square matrix is a value in R assigned to the matrix, it characterizes matrices which are invertible (det 0) and is related to the volume of a parallelpiped described

### Matrix Inverse and Determinants

DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### Lecture 11. Shuanglin Shao. October 2nd and 7th, 2013

Lecture 11 Shuanglin Shao October 2nd and 7th, 2013 Matrix determinants: addition. Determinants: multiplication. Adjoint of a matrix. Cramer s rule to solve a linear system. Recall that from the previous

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### 2.5 Elementary Row Operations and the Determinant

2.5 Elementary Row Operations and the Determinant Recall: Let A be a 2 2 matrtix : A = a b. The determinant of A, denoted by det(a) c d or A, is the number ad bc. So for example if A = 2 4, det(a) = 2(5)

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### Chapter 8. Matrices II: inverses. 8.1 What is an inverse?

Chapter 8 Matrices II: inverses We have learnt how to add subtract and multiply matrices but we have not defined division. The reason is that in general it cannot always be defined. In this chapter, we

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### NON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that

NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called

### Math 315: Linear Algebra Solutions to Midterm Exam I

Math 35: Linear Algebra s to Midterm Exam I # Consider the following two systems of linear equations (I) ax + by = k cx + dy = l (II) ax + by = 0 cx + dy = 0 (a) Prove: If x = x, y = y and x = x 2, y =

### MATH36001 Background Material 2015

MATH3600 Background Material 205 Matrix Algebra Matrices and Vectors An ordered array of mn elements a ij (i =,, m; j =,, n) written in the form a a 2 a n A = a 2 a 22 a 2n a m a m2 a mn is said to be

### = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### Topic 1: Matrices and Systems of Linear Equations.

Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### Determinants. Chapter Properties of the Determinant

Chapter 4 Determinants Chapter 3 entailed a discussion of linear transformations and how to identify them with matrices. When we study a particular linear transformation we would like its matrix representation

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### The Inverse of a Matrix

The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices

Matrices 2. Solving Square Systems of Linear Equations; Inverse Matrices Solving square systems of linear equations; inverse matrices. Linear algebra is essentially about solving systems of linear equations,

### Sergei Silvestrov, Christopher Engström, Karl Lundengård, Johan Richter, Jonas Österberg. November 13, 2014

Sergei Silvestrov,, Karl Lundengård, Johan Richter, Jonas Österberg November 13, 2014 Analysis Todays lecture: Course overview. Repetition of matrices elementary operations. Repetition of solvability of

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL

SOLUTIONS OF THEORETICAL EXERCISES selected from INTRODUCTORY LINEAR ALGEBRA WITH APPLICATIONS B. KOLMAN, D. R. HILL Eighth Edition, Prentice Hall, 2005. Dr. Grigore CĂLUGĂREANU Department of Mathematics

### ( % . This matrix consists of \$ 4 5 " 5' the coefficients of the variables as they appear in the original system. The augmented 3 " 2 2 # 2 " 3 4&

Matrices define matrix We will use matrices to help us solve systems of equations. A matrix is a rectangular array of numbers enclosed in parentheses or brackets. In linear algebra, matrices are important

### Chapter 5. Matrices. 5.1 Inverses, Part 1

Chapter 5 Matrices The classification result at the end of the previous chapter says that any finite-dimensional vector space looks like a space of column vectors. In the next couple of chapters we re

### Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Chapter 1. Determinants

Chapter 1. Determinants This material is in Chapter 2 of Anton & Rorres (or most of it is there). See also section 3.5 of that book. 1.1 Introductory remarks The determinant of a square matrix A is a number

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v

### 1. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither.

Math Exam - Practice Problem Solutions. For each of the following matrices, determine whether it is in row echelon form, reduced row echelon form, or neither. (a) 5 (c) Since each row has a leading that

### 1 Determinants and the Solvability of Linear Systems

1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### EC9A0: Pre-sessional Advanced Mathematics Course

University of Warwick, EC9A0: Pre-sessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Pre-sessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond

### In this leaflet we explain what is meant by an inverse matrix and how it is calculated.

5.5 Introduction The inverse of a matrix In this leaflet we explain what is meant by an inverse matrix and how it is calculated. 1. The inverse of a matrix The inverse of a square n n matrix A, is another

### Unit 18 Determinants

Unit 18 Determinants Every square matrix has a number associated with it, called its determinant. In this section, we determine how to calculate this number, and also look at some of the properties of

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### Chapter 1 - Matrices & Determinants

Chapter 1 - Matrices & Determinants Arthur Cayley (August 16, 1821 - January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed

### Chapter 4: Binary Operations and Relations

c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

### Problems. Universidad San Pablo - CEU. Mathematical Fundaments of Biomedical Engineering 1. Author: First Year Biomedical Engineering

Universidad San Pablo - CEU Mathematical Fundaments of Biomedical Engineering 1 Problems Author: First Year Biomedical Engineering Supervisor: Carlos Oscar S. Sorzano September 15, 013 1 Chapter 3 Lay,

### Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants

Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview

### Elementary Row Operations and Matrix Multiplication

Contents 1 Elementary Row Operations and Matrix Multiplication 1.1 Theorem (Row Operations using Matrix Multiplication) 2 Inverses of Elementary Row Operation Matrices 2.1 Theorem (Inverses of Elementary

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### Preliminaries of linear algebra

Preliminaries of linear algebra (for the Automatic Control course) Matteo Rubagotti March 3, 2011 This note sums up the preliminary definitions and concepts of linear algebra needed for the resolution

### 9 Matrices, determinants, inverse matrix, Cramer s Rule

AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### APPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the co-factor matrix [A ij ] of A.

APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the co-factor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj

### Solution. Area(OABC) = Area(OAB) + Area(OBC) = 1 2 det( [ 5 2 1 2. Question 2. Let A = (a) Calculate the nullspace of the matrix A.

Solutions to Math 30 Take-home prelim Question. Find the area of the quadrilateral OABC on the figure below, coordinates given in brackets. [See pp. 60 63 of the book.] y C(, 4) B(, ) A(5, ) O x Area(OABC)

### GRA6035 Mathematics. Eivind Eriksen and Trond S. Gustavsen. Department of Economics

GRA635 Mathematics Eivind Eriksen and Trond S. Gustavsen Department of Economics c Eivind Eriksen, Trond S. Gustavsen. Edition. Edition Students enrolled in the course GRA635 Mathematics for the academic

### Matrices, Determinants and Linear Systems

September 21, 2014 Matrices A matrix A m n is an array of numbers in rows and columns a 11 a 12 a 1n r 1 a 21 a 22 a 2n r 2....... a m1 a m2 a mn r m c 1 c 2 c n We say that the dimension of A is m n (we

### The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices

The Laplace Expansion Theorem: Computing the Determinants and Inverses of Matrices David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created:

### Using row reduction to calculate the inverse and the determinant of a square matrix

Using row reduction to calculate the inverse and the determinant of a square matrix Notes for MATH 0290 Honors by Prof. Anna Vainchtein 1 Inverse of a square matrix An n n square matrix A is called invertible

### DETERMINANTS TERRY A. LORING

DETERMINANTS TERRY A. LORING 1. Determinants: a Row Operation By-Product The determinant is best understood in terms of row operations, in my opinion. Most books start by defining the determinant via formulas

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### Chapter 5. Determinants. Po-Ning Chen, Professor. Department of Electrical and Computer Engineering. National Chiao Tung University

Chapter 5 Determinants Po-Ning Chen, Professor Department of Electrical and Computer Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. 5.1 The properties of determinants 5-1 Recall

### Practice Math 110 Final. Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16.

Practice Math 110 Final Instructions: Work all of problems 1 through 5, and work any 5 of problems 10 through 16. 1. Let A = 3 1 1 3 3 2. 6 6 5 a. Use Gauss elimination to reduce A to an upper triangular

### Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.

Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### Mathematics Notes for Class 12 chapter 3. Matrices

1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor

### Additional Topics in Linear Algebra Supplementary Material for Math 540. Joseph H. Silverman

Additional Topics in Linear Algebra Supplementary Material for Math 540 Joseph H Silverman E-mail address: jhs@mathbrownedu Mathematics Department, Box 1917 Brown University, Providence, RI 02912 USA Contents

### MATHEMATICS FOR ENGINEERS BASIC MATRIX THEORY TUTORIAL 2

MATHEMATICS FO ENGINEES BASIC MATIX THEOY TUTOIAL This is the second of two tutorials on matrix theory. On completion you should be able to do the following. Explain the general method for solving simultaneous

### Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved.

Matrix Algebra, Class Notes (part 1) by Hrishikesh D. Vinod Copyright 1998 by Prof. H. D. Vinod, Fordham University, New York. All rights reserved. 1 Sum, Product and Transpose of Matrices. If a ij with

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### MA 242 LINEAR ALGEBRA C1, Solutions to Second Midterm Exam

MA 4 LINEAR ALGEBRA C, Solutions to Second Midterm Exam Prof. Nikola Popovic, November 9, 6, 9:3am - :5am Problem (5 points). Let the matrix A be given by 5 6 5 4 5 (a) Find the inverse A of A, if it exists.

### 1 Eigenvalues and Eigenvectors

Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x

### Homework: 2.1 (page 56): 7, 9, 13, 15, 17, 25, 27, 35, 37, 41, 46, 49, 67

Chapter Matrices Operations with Matrices Homework: (page 56):, 9, 3, 5,, 5,, 35, 3, 4, 46, 49, 6 Main points in this section: We define a few concept regarding matrices This would include addition of

### Matrices: 2.3 The Inverse of Matrices

September 4 Goals Define inverse of a matrix. Point out that not every matrix A has an inverse. Discuss uniqueness of inverse of a matrix A. Discuss methods of computing inverses, particularly by row operations.

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### The Cayley Hamilton Theorem

The Cayley Hamilton Theorem Attila Máté Brooklyn College of the City University of New York March 23, 2016 Contents 1 Introduction 1 1.1 A multivariate polynomial zero on all integers is identically zero............

### 1 Spherical Kinematics

ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

### Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

### Diagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions

Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential

### Solving Linear Systems, Continued and The Inverse of a Matrix

, Continued and The of a Matrix Calculus III Summer 2013, Session II Monday, July 15, 2013 Agenda 1. The rank of a matrix 2. The inverse of a square matrix Gaussian Gaussian solves a linear system by reducing

### 26. Determinants I. 1. Prehistory

26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent

### Derivatives of Matrix Functions and their Norms.

Derivatives of Matrix Functions and their Norms. Priyanka Grover Indian Statistical Institute, Delhi India February 24, 2011 1 / 41 Notations H : n-dimensional complex Hilbert space. L (H) : the space