Matrix Algebra and Applications


 Marylou Owen
 2 years ago
 Views:
Transcription
1 Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49
2 EC2040 Topic 2  Matrices and Matrix Algebra Reading 1 Chapters 4 and 5 of CW 2 Chapters 11, 12 and 13 of PR Plan 1 Matrices and Matrix Algebra 2 Transpose, Inverse, and Determinant of a Matrix 3 Solutions to Systems of Linear Equations Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 2 / 49
3 Matrices and System of Equations A matrix is an array of numbers. Some examples are, [ ] A =, B = 5 10, C = Notation: We shall use a capital letter to denote a matrix and the corresponding small letter to denote individual elements of a matrix. The element in the (2, 1) position (2nd [ row, 1st column) ] of A will be a11 a denoted as a 21. So, from above, A = 12. a 21 a 22 The number of rows and columns in a matrix is also called its order or dimension. For instance, the matrix A has order Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 3 / 49
4 Special Types of Matrices Square matrix which has the same number of rows as columns (order n n) Row matrix (order 1 n) Column matrix (order n 1). A vector is a matrix having either a single row or a single column. Two special types of square matrices are the null matrix (all entries zero) and the identity matrix which has 1 along the diagonal and zeros everywhere else: A square matrix where the only nonzero entries are on the diagonal is also called a diagonal matrix. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 4 / 49
5 Matrices in Economics One example of a matrix in economics is the ISLM macro model. 1 Y = C + I + Ḡ C = a + b(1 t)y I = e lr M = ky hr (resource constraint) (consumption function) (investment function) (money demand) We can write the above system in matrix notation as: Y b(1 t) C l I = k 0 0 h R Question: what happens to consumption when we increase the money supply (expansionary monetary policy). 1 Recall, IS (the first three equations) and LM gives AD, where Ḡ and M are government policies. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 5 / 49 Ḡ a e M
6 Arithmetical Operations on Matrices Addition and Subtraction of matrices is very easy but can be done only when all the matrices have the same dimension. Thus, A m n + B p q is defined only when m = p and n = q. Example: [ ] =A [ ] =B 2 2 = [ ] = [ That is, m = p = n = q = 2. [ ] c a11 a 11 Clearly, 12 and c a 21 a 12 cannot be added together. 22 c =A =C 3 1 ] Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 6 / 49
7 Arithmetical Operations on Matrices Formal Definition: Suppose that we have two matrices A = [a ij ] m n and B = [b ij ] m n. Then, the matrix A + B is simply [a ij + b ij ] m n. E.g., for the m = n = 2 case, [ ] [ ] [ ] a11 a 12 b11 b 12 a11 + b 11 a 12 + b 12 a 21 a 22 =A b 21 b 22 B 2 2 = Similarly, the matrix A B is simply [a ij b ij ] m n. a 21 + b 21 a 22 + b 22 This process easily extends to the case when we have many matrices. For instance  if we have three matrices A, B, C of the same dimension  then A + B + C is the matrix formed by adding the corresponding (i, j)th entries of A, B and C. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 7 / 49
8 Scalar Multiplication Scalar Multiplication by a number c is just multiplying each element of the matrix by the number c. 2 1 For instance if A = and c = 5/2 then c A is the 2 2 matrix 5 5/2 15/2 45/ / Note that scalar multiplication applies to any matrix. We can combine the operations of scalar multiplication and addition: for instance, you should be able to say what 5A + 3B means if A and B have the same dimension. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 8 / 49
9 Matrix Multiplication Matrix Multiplication is somewhat more complicated (but all it really requires is concentration). Suppose that we have two matrices A m n and B p q. The product AB is defined only when n = p, that is when the number of columns in A equals the number of rows in B. 1 This means that the product BA is defined only when q = m. 2 It is possible that AB is defined but BA is not defined. 3 Furthermore, even if AB and BA are both defined it is possible that they do not give the same matrix. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 9 / 49
10 Matrix Multiplication The bottom line is, the order of operation is important in matrix multiplication. Mathematically, we say that matrix multiplication is not commutative (whereas the scalar case is). All this is very different from ordinary multiplication. Take the previous example: [ ] 4 3 A 2 2 = and B = What are AB and BA? [ ] Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 10 / 49
11 Matrix Multiplication for the 2 by 2 case First, AB is, [ ] [ ] AB = [ ] 4 (1) + 3 (5) 4 (2) + 3 (6) = = 1 (1) + 2 (5) 1 (2) + 2 (6) However, BA is, [ ] [ ] BA = [ ] 1 (4) + 2 (1) 1 (3) + 2 (2) = = 5 (4) + 6 (1) 5 (3) + 6 (2) So, AB is not BA. [ [ Again, think about this versus the scalar case. If a = 5 and b = 6, ab = ba = 30. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 11 / 49 ] ]
12 Formal Definition of Matrix Multiplication Suppose we have two matrices A = [a ij ] m n and B = [b ij ] n p. The product AB is a m p matrix [c ij ] m p where, c ij = a i1 b 1j + a i2 b 2j a in b nj. Loosely speaking, c ij is the product of the ith row of A and the jth column of B. Again, for the 2 by 2 case, we have: [ ] [ ] a11 a AB = 12 b11 b 12 a 21 a 22 b 21 b 22 [ ] a11 (b = 11 ) + a 12 (b 21 ) a 11 (b 11 ) + a 12 (b 21 ) a 21 (b 12 ) + a 22 (b 22 ) a 21 (b 12 ) + a 22 (b 22 ) [ ] c11 c = 12 C c 21 c 22 Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 12 / 49
13 Example Suppose A = possible and B = [ ] and AB = C, if Since A has order 3 2 and B has order 2 2, it follows that C = AB is defined and the product is a matrix of dimension 3 2. The individual elements of C (e.g., c 11 = a 11 b 11 + a 12 b 21 ) are: c 11 = (1 7) + (2 9) = 25, c 12 = (1 8) + (2 10) = 28, c 21 = (3 7) + (4 9) = 57, c 22 = (3 8) + (4 10) = 64, c 31 = (5 7) + (6 9) = 89, c 32 = = 100. Hence, C = c 11 c 12 c 21 c 22 c 31 c 32 = Note that BA is not defined as elements a 31 = 5 and a 32 = 6 cannot be matched with elements of B.. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 13 / 49
14 Null and Identity Matrix Multiplication The null and identity matrix cases are trivial. The identity matrix plays the role of 1 in the scalar case. Scalar case: Matrix case: [ ] [ ] = 3 1 = 3 [ 4 (1) + 3 (0) 4 (0) + 3 (1) 1 (1) + 2 (0) 1 (0) + 2 (1) ] = [ The null matrix plays the role of zero. As with 3 0 = 0, so, [ ] [ ] [ ] [ (0) + 3 (0) 4 (0) + 3 (0) 0 0 = = (0) + 2 (0) 1 (0) + 2 (0) 0 0 ] ] Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 14 / 49
15 Recap: Rules for Scalars versus Matrices Scalar cases: 1 Commutative; a + b = b + a 2 Associative; (a + b) + c = a + (b + c) and (ab) c = a (bc) 3 Distributive; a (b + c) = ab + ac and (b + c) a = ba + ca Matrix cases: 1 Addition Commutative law; A + B = B + A Associative law; (A + B) + C = A + (B + C ) 2 Multiplication Associative law; (AB) C = A (BC ) Distributive law; A (B + C ) = AB + AC and (B + C ) A = BA + CA Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 15 / 49
16 Transpose of a Matrix The transpose of a matrix A is formed by interchanging the rows and columns of A. If A has order m n, then it s transpose has dimension n m. The transpose of A is generally denoted as A T and sometimes A. If A = A T, then we say that A is a symmetric matrix. [ A ] symmetric 1 3 matrix must be a square matrix (e.g., A = A T = ). 3 1 Transpose of a square matrix: [ ] [ ] a11 a A = 12 A T a11 a = 21 a 21 a 22 a 12 a 22 Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 16 / 49
17 Properties of the Transpose The transpose operation has the following properties. 1 (A T ) T = A [ ] [ (A + B) T = A T + B T. Suppose A = A 1 2 T = 3 2 [ ] [ ] and B = B 5 6 T =. Try this at home (AB) T = B T A T More generally, if we have nmatrices A 1, A 2,..., A n then ] (A 1 A 2... A n ) T = A T n A T n 1... A T 2 A T 1 Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 17 / 49
18 The Inverse of a Square Matrix For a square matrix, A n n, the following is always true: AI n = I n A = A If we can find a matrix B n n satisfying AB = BA = I n, then we say that B is the inverse matrix of A. The inverse matrix of A is denoted A 1. Not all square matrices have inverses. 1 A matrix which has an inverse is called nonsingular. 2 A matrix which does not have an inverse is called singular. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 18 / 49
19 The Determinant of a Square Matrix We denote the determinant of A, as A. The determinant of a 1 1 matrix is trivial: it is the number itself. [ ] a11 a The determinant of a 2 2 matrix A = 12 is, a 21 a 22 A = a 11 a 22 a 12 a 21 The procedure for bigger matrices is more complicated. Let A be a n n matrix. Let M ij denote the determinant of the matrix derived from A by deleting the ith row and jth column. Let C ij = ( 1) i+j M ij. We refer to M ij as a minor and C ij as a cofactor. Note: A square matrix A has an inverse if and only if its determinant is nonzero (is nonsingular). Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 19 / 49
20 The Determinant of a Square matrix Pick any row or column in the matrix. Suppose we pick the ith row. Then, the determinant of the matrix A is: A = a i1 C i1 + a i2 C i a in C in If we pick the a column, say the jth one, then the determinant is computed as: A = a 1j C 1j + a 2j C 2j a nj C nj Observe that we could have picked any row or column: we will always get the same answer. For practical considerations, we try to pick out a row or column which has the maximum number of zeros. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 20 / 49
21 The Determinant of a Square matrix, 2 by 2 case Suppose we have the following matrix: A = A = (1 4) (2 3) = 2. [ ]. We know Although it s a little trivial, we can relate that to the technique for larger matrices, as a first step. We have, A = (a 11 C 11 ) + (a 12 C 12 ) = [ a 11 ( 1) 1+1 M 11 ] + [ a12 ( 1) 1+2 M 12 ] = a 11 M 11 a 12 M 12 ; M 11 = 4 and M 12 = 3 = a 11 4 a 12 3 = (1 4) (2 3) = 2 Why is this all so easy? In the 2 by 2 case, the M ij s are scalars. If we have a 3 by 3 case, the M ij are 2 by 2 matrices, which we have to work out the determinant of first etc... Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 21 / 49
22 3 by 3 Example Suppose we have the following 3 by 3 matrix, A = Let us compute the determinant by computing the cofactors of the first row. We have, C ij = ( 1) i+j M ij, so, C 11 = ( 1) 1+1 M 11 = ( 1) } {{ } M 11 But we know that M 11 = (8 6) (2 9) = = 30. We conclude that C 11 = 30.. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 22 / 49
23 3 by 3 Example Continued We can do this for the other 2 elements of the row. C 12 = ( 1) = 6 } {{ } =M 12 C 13 = ( 1) = 18 We find, Hence, the determinant, call it for short, is, = a 11 C 11 + a 12 C 12 + a 13 C 13 = a 11 M 11 a 21 M 21 + a 13 M 13 = [1 ( 30)] + (2 6) + (3 18) = 36 We could have chosen any other row or column. If we choose the third column, then we have: C 13 = 18, C 23 = 6, C 33 = 6 and therefore = (3 18) + (6 6) + [9 ( 6)] = 36. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 23 / 49
24 Properties of the Determinant There are some useful properties of determinants (that we won t prove) which can simplify the computation of the determinant significantly. 1 If a row or column of A is multiplied by c, then the determinant of the new matrix is c A. 2 Multiplying a row (column) by a nonzero constant and adding it to another row (column) has no effect on the determinant. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 24 / 49
25 Extended Example with Common Factors Common factors basically involves taking things outside the brackets, as we usually do, except we can use the properties of the determinant Consider the same matrix as before, A = Note that the second row can be written as 2 [ ] = [ ]. We know that multiplying a row by a constant leads the determinant to change by the same factor. It follows, A = Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 25 / 49
26 Example Continued Note that column three also has a common factor of 3, i.e., = It follows that, A = (2 3) = There are now no common factors; however we can now make use of the row and column operations. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 26 / 49
27 Example Continued Since the determinant is not changed by multiplying a row (column) by a constant and adding it to another row (column) we can use this to make some entries in a row or column zeros. Multiplying the first row by 1 and adding it to the second gives, A = What did we do? Well, we started with and we noted, T T T ( 1) = 1, which is the new middle row of A. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 27 / 49
28 Example Continued Similarly, multiplying the first row by 3 and adding it to the third row gives: A = Following the same idea as before, we start out with A = and we note, T T T ( 3) = 2, which is the new bottom row of A. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 28 / 49
29 Example Continued Why did we do all of this? Well, A = is not that easy to compute However, A = is easy to compute, as we can take the cofactors of the third column. In that case, we only have to compute one cofactor since two entries in the column are zero. Therefore, we have, A = 6 [ 1 ( 1) ] = 6 6 = 36 Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 29 / 49
30 Computing the inverse using the determinant We observed before that we can find the inverse of a square matrix A if and only if A = 0. When the determinant exists, it can be shown that the inverse is given by, A 1 = 1 A [c ij] T where [c ij ] is the matrix of cofactors of A. In words, what we do is the following: 1 Replace each element of A by its cofactor. 2 Take the transpose of the matrix. 3 Scalar multiply this matrix by 1/ A. Note: the following is also used, adj (A) [c ij ] T ; it is called the adjunct. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 30 / 49
31 A 2 by 2 Example of Matrix Inversion Consider the following 2 by 2 matrix A = [ We know that A = (3 0) (1 2) = 2. inverse must exist. ]. Since A = 0, an The cofactor matrix (notice how [ easy it is ] in this case, [ again due to ] zeros) is the following; c ij = [c 2 3 ij ] T = 1 3 [ ] [ ] Then, A 1 = 1 A [c ij] T = = Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 31 / 49
32 The Inverse of a Square Matrix Consider the following 2 by 2 matrix: A = matrix have an inverse? [ 1 λ 3 4 ]. Does this We know an inverse only exists if the matrix is nonsingular. to hold, we require, A = 0. For this We know that if A = (1 4) (3 λ) = 0 the matrix is singular. That is, we can t find the inverse of A if λ = 4/3. Finding a singularity tends to become a problem is we have very big matrices (say we have a big model) which is relatively sparse (that is, has lots of zeros in it). Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 32 / 49
33 3 by 3 Example Let us compute the inverse of the matrix that we have seen before: A = We already know the following: C 11 = 30, C 12 = 6, C 13 = 18, C 23 = 6, C 33 = 6 A = 36 Recall we can use C 1j s or C i3 s to get here. The remaining cofactors are C 21 = 6, C 22 = 12, C 31 = 6, C 32 = 6. Try this at home. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 33 / 49
34 Matrix Inversion Example Continued Here the matrix of cofactors is, [c ij ] = The transpose is, [c ij ] T = The inverse matrix is, A 1 = 1 A [c ij] T = A = as Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 34 / 49
35 Cramer s Rule Now suppose we have a big linear model. In it s most general specification it is written as a system of linear equations: a 11 x 1 + a 12 x a 1n x n = b =. a n1 x 1 + a n2 x a nn x n = b n This is system of n linear equations in n unknowns. It can be represented compactly in matrix notation as Ax = b where, a a 1n x 1 b 1 A =..., x =., b =.. a n1... a nn x n b n Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 35 / 49
36 Cramer s Rule We note that providing A = 0 (that is, A has an inverse) we can solve for x by just multiplying both sides of the equation Ax = b by A 1 : (A 1 )Ax = (A 1 A)x = I n x = x = A 1 b Cramer s rule is just an alternative formula for this; the advantage of the rule is that we can find each x i individually; this may be useful if we don t want to find the value of all x i. Cramer s Rule states that: x i = A b,i A Here A b,i is the matrix where the ith column of A is replaced by the column vector b. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 36 / 49
37 Examples of solving for a specific variable Consider the system hence the system is solvable. A w y z x = We can compute w independently of y and z as, w = = 48 A 80 = 3 5 Computation of y and z is left as an exercise. b. Here, A = 80; Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 37 / 49
38 Why does Cramer s Rule work? Cramer s Rule is equivalent to finding the inverse of A. To see this more formally, consider the system, a 11 a 12 a 13 w b 1 a 21 a 22 a 33 y = b 2. a 31 a 32 a 33 z b 3 Now note that the inverse of A can be written as, A 1 = 1 A where C ij is the cofactor of a ij. C 11 C 21 C 31 C 12 C 22 C 32 C 13 C 23 C 33 Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 38 / 49
39 Why does Cramer s Rule work? Hence, we have, w y = A 1 b = 1 A z b 1 C 11 + b 2 C 21 + b 3 C 31 b 1 C 12 + b 2 C 22 + b 3 C 32 b 1 C 13 + b 2 C 23 + b 3 C 33 Now note that b 1 C 11 + b 2 C 21 + b 3 C 31 is nothing but the determinant of the matrix where the first column of A has been replaced by the vector b. Similarly, b 1 C 12 + b 2 C 22 + b 3 C 32 is the determinant of the matrix where the second column of A has been replaced by the vector b. You can check that a similar observation holds for b 1 C 13 + b 2 C 23 + b 3 C 33. This shows that the inverse approach and Cramer s rule are identical for a system of three linear equations but this approach can be easily extended. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 39 / 49
40 A Demand and Supply Example Consider a linear model of demand and supply (of icecreams). 1 Demand for icecreams depends on the price of icecreams, the income of individuals and the temperature. 2 The supply of icecreams also depends on how much the icecream company can charge for it s product and how hot it is. q d = γm bp + αt q s = δp ɛt + d If income goes up, demand goes up. If the temperature goes up, demand goes up but supply goes down (some of the icecream melts before it is sold). If price goes up, demand falls and supply rises. In equilibrium, q d = q s. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 40 / 49
41 A Demand and Supply Example It is very easy to solve this model. The solution for the price is, p γm+(α+ɛ)t d = δ+b. Higher temperatures result in higher prices as people eat more icecreams and icecreams melt, reducing supply. We can also draw the model in (p, q) space and shift the curves. More importantly, we can also represent the model in matrix form. [ ] [ ] [ ] b 1 p γm + αt = δ 1 q d ɛt A Now we need A = b + δ = 0. If we look at p, that makes a lot of economic sense. We can t have an infinite price for icecream. Again, we can also use Cramer s rule to solve for shocks to income or temperature. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 41 / 49
42 Demand and Supply The supply and demand example is trivial. However, we might imagine that the entire economy (not just one market) consists of many demand and supply schedules (what we called a big linear model). The demand and supply schedules will probably display some form of interdependence. That is, consider an economy with icecreams and apples. If the temperature goes up, the demand for apples might drop. Why? Because, with given income, the demand for icecream rises. If this is the case, our economy consists of an entire vector of prices and a vector of quantities. We want to know the price and quantities in equilibrium. That turns out to be very complicated. Having a grasp of matrix algebra is then very useful as we want to know if an equilibrium exists. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 42 / 49
43 Returning to the example at the very beginning Consider the macroeconomic model we wrote down at the beginning. C = a + b(1 t)y I = e lr G = Ḡ L = ky hr M = M We want to solve this model to determine the endogenous variables (Y, C, I, R) in terms of the exogenous variables (Ḡ, a, b, t, e, k, h, l, M). We can use all the new tools we have developed to solve the model. We ll focus on solving for consumption as a function of monetary and fiscal policy. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 43 / 49
44 Macro Model Example The equilibrium in this system is determined by the conditions: Y = C + I + Ḡ C = a + b(1 t)y I = e lr M = ky hr We can write the above system in matrix notation as: Y b(1 t) C l I = k 0 0 h R The first thing to do is to compute A which is needed no matter which approach we use to solve this system. We choose one of the rows or columns with the maximum number of zeros since that will simplify the computation. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 44 / 49 Ḡ a e M
45 Example Continued Suppose we choose row 2. We know, = 4 j=1 a 2j C 2j. Then, = b(1 t) C 21 + C 22, as a 22 = 1 and a 23 = a 24 = 0. } {{ } =a21 Also C ij = ( 1) i+j M ij so = 4 j=1 a 2j ( 1) 2+j M 2j. However, it is not as easy as before because the M 2j s are 3 by 3 matrices. We only have two of them, in this case. C 21 = ( 1) l C 22 = ( 1) h l k 0 h Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 45 / 49
46 Example Continued Expanding the 3 3 determinant in the expression for C 21 about the first column (two zero elements), we get, C 21 = [( 1)(1 ( h) 0 l)] = h For C 22 we can expand the 3 3 determinant about the first column (one zero element): C 22 = ( 1) l 0 h + ( 1)3+1 k l = h kl Hence the determinant of the original 4 4 is, = bh(1 t) h kl = h(1 b(1 t)) kl Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 46 / 49
47 Example Continued We can now compute the value of any of the endogenous variables using Cramer s Rule. Suppose we want to compute C. We replace the second column with [ Ḡ a e M ] T. Then, C = 1 A 1 Ḡ 1 0 b(1 t) a e 1 l k M 0 h Computing the determinant of the 4 4 matrix by expanding around the second row, we have = b(1 t) ( 1) 2+1 Ḡ 1 0 e 1 l M 0 h + a ( 1) l k 0 h Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 47 / 49
48 Example Continued The second 3 3 determinant above has already been found. Expanding this about the first 3 3 around the first row, we have, Ḡ 1 0 e 1 l M 0 h = Ḡ ( 1)2 [1 ( h) 0 l] + ( 1) This is (hḡ + he + Ml). ( 1) 3 [e ( h) M l] Hence, = b(1 t) (hḡ + he + Ml) + a (h + kl) = [b(1 t)(hḡ + he + Ml) + a(h + kl)]. It follows that C = b(1 t)(hḡ + he + Ml) + a(h + kl) h(1 b(1 t)) + kl This is all exogenous; and Ḡ, M are policy variables. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 48 / 49
49 Roundup You should now be able to do the following: 1 Add/subtract/multiply matrices. 2 Find the inverse of a matrix and it s determinant (which is basically matrix manipulation). 3 Apply these techniques to standard models and use Cramer s rule to solve for endogenous variables as functions of exogenous variables. Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 49 / 49
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
More information9 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:
More information(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
More information= [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
More informationMATRIX 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 +
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More information( % . 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
More informationMATH 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 mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationHelpsheet. 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
More informationMATH 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
More informationInverses 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
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationMATRIX 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
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationMathematics 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
More informationHomework: 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
More informationWe 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
More informationDETERMINANTS. 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
More informationCofactor 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
More informationThe 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
More informationThe 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
More information1.5 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:
More informationDeterminants. 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.
More informationAPPLICATIONS OF MATRICES. Adj A is nothing but the transpose of the cofactor matrix [A ij ] of A.
APPLICATIONS OF MATRICES ADJOINT: Let A = [a ij ] be a square matrix of order n. Let Aij be the cofactor of a ij. Then the n th order matrix [A ij ] T is called the adjoint of A. It is denoted by adj
More informationTopic 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
More informationSolution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationB such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix
Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationIntroduction 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
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationChapter 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
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationMatrix 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
More informationChapter 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
More informationSYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89. by Joseph Collison
SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More informationMatrices, 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
More informationrow row row 4
13 Matrices The following notes came from Foundation mathematics (MATH 123) Although matrices are not part of what would normally be considered foundation mathematics, they are one of the first topics
More information1 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
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationThe 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, highdimensional
More informationIn 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
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationMath 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 =
More information13 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 threespace, we write a vector in terms
More information1 Vector Spaces and Matrix Notation
1 Vector Spaces and Matrix Notation De nition 1 A matrix: is rectangular array of numbers with n rows and m columns. 1 1 1 a11 a Example 1 a. b. c. 1 0 0 a 1 a The rst is square with n = and m = ; the
More informationMATH36001 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
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationMatrices: 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.
More informationLecture Notes: Matrix Inverse. 1 Inverse Definition
Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,
More informationMatrices Summary. To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers.
Matrices Summary To transpose a matrix write the rows as columns. Academic Skills Advice For example: 2 1 A = [ 1 2 1 0 0 9] A T = 4 2 2 1 2 1 1 0 4 0 9 2 To add or subtract matrices they must be the same
More informationMath 2331 Linear Algebra
2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University
More informationMATH 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
More informationSergei 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
More information2.1: MATRIX OPERATIONS
.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More informationEC9A0: Presessional Advanced Mathematics Course
University of Warwick, EC9A0: Presessional Advanced Mathematics Course Peter J. Hammond & Pablo F. Beker 1 of 55 EC9A0: Presessional Advanced Mathematics Course Slides 1: Matrix Algebra Peter J. Hammond
More informationCalculus 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.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationChapter 7. Matrices. Definition. An m n matrix is an array of numbers set out in m rows and n columns. Examples. ( 1 1 5 2 0 6
Chapter 7 Matrices Definition An m n matrix is an array of numbers set out in m rows and n columns Examples (i ( 1 1 5 2 0 6 has 2 rows and 3 columns and so it is a 2 3 matrix (ii 1 0 7 1 2 3 3 1 is a
More informationMatrices Worksheet. Adding the results together, using the matrices, gives
Matrices Worksheet This worksheet is designed to help you increase your confidence in handling MATRICES. This worksheet contains both theory and exercises which cover. Introduction. Order, Addition and
More information2.5 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)
More informationInverses. Stephen Boyd. EE103 Stanford University. October 27, 2015
Inverses Stephen Boyd EE103 Stanford University October 27, 2015 Outline Left and right inverses Inverse Solving linear equations Examples Pseudoinverse Left and right inverses 2 Left inverses a number
More information2.1: 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
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More information8 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
More informationMatrix 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
More informationA 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
More informationLecture 10: Invertible matrices. Finding the inverse of a matrix
Lecture 10: Invertible matrices. Finding the inverse of a matrix Danny W. Crytser April 11, 2014 Today s lecture Today we will Today s lecture Today we will 1 Single out a class of especially nice matrices
More informationDiagonal, 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
More informationSECTION 8.3: THE INVERSE OF A SQUARE MATRIX
(Section 8.3: The Inverse of a Square Matrix) 8.47 SECTION 8.3: THE INVERSE OF A SQUARE MATRIX PART A: (REVIEW) THE INVERSE OF A REAL NUMBER If a is a nonzero real number, then aa 1 = a 1 a = 1. a 1, or
More informationMathematics 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
More informationL12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014
L12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.  Morpheus Primary concepts:
More informationSCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING. Self Study Course
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I ENGINEERING Self Study Course MODULE 17 MATRICES II Module Topics 1. Inverse of matrix using cofactors 2. Sets of linear equations 3. Solution of sets of linear
More information1 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
More informationLecture 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
More informationLinear 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 matrixvector notation as 4 {{ A x x x {{ x = 6 {{ b General
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
More informationIntroduction to Modular Arithmetic, the rings Z 6 and Z 7
Introduction to Modular Arithmetic, the rings Z 6 and Z 7 The main objective of this discussion is to learn modular arithmetic. We do this by building two systems using modular arithmetic and then by solving
More information1 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 GaussJordan reduction and the Reduced
More informationEconomics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics
Economics 102C: Advanced Topics in Econometrics 2  Tooling Up: The Basics Michael Best Spring 2015 Outline The Evaluation Problem Matrix Algebra Matrix Calculus OLS With Matrices The Evaluation Problem:
More informationMatrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.
2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationMatrix Solution of Equations
Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to
More informationSolving a System of Equations
11 Solving a System of Equations 111 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
More informationFurther Maths Matrix Summary
Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number
More informationSolving 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
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationLinear 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
More informationMATHEMATICS 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
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationUnit 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
More informationLINEAR ALGEBRA. September 23, 2010
LINEAR ALGEBRA September 3, 00 Contents 0. LUdecomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationDiagonalisation. 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
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More information