Matrices: 2.2 Properties of Matrices


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2 Goals We will describe properties of matrices with respect to addition, scalar multiplications and matrix multiplication and others. Among what we will see 1. Matrix multiplication does not commute. That means, not always AB = BA. 2. We will define transpose A T of a matrix A and discuss its properties.
3 Algebra of Matrices Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Let A,B,C be m n matrices and c,d be scalars. Then, A+B = B +A Commutativity of addition A+(B +C) = (A+B)+C Associativity of addition (cd)a = c(da) Associativity of scalar multiplication c(a+b) = ca+cb a Distributive property (c +d)a = ca+da a Distributive property These seem obvious, expected and are easy to prove.
4 Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Zero The m n matrix with all entries zero is denoted by O mn. For a matrix A of size m n and a scalar c, we have A+O mn = A (This property is stated as:o mn is the additive identity in the set of all m n matrices.) A+( A) = O mn. (This property is stated as: A is the additive inverse of A.) ca = O mn = c = 0 or A = O mn. Remark. So far, it appears that matrices behave like real numbers. Reading Assignment: Example 2 from 2.2 of the textbook.
5 Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Properties of Matrix Multiplication Let A,B,C be matrices and c is a constant. Assume all the matrix products below are defined. Then A(BC) = (AB)C A(B +C) = AB +AC (A+B)C = AC +BC c(ab) = (ca)b = A(cB) Associativity Matrix Product Distributive Property Distributive Property Proofs would be routine checking, which we would skip. Reading Assignment: Example 3, 4, 5 from 2.2 of the textbook.
6 Definition Preview Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions For a positive integer, I n would denote the square matrix of order n whose main diagonal (left to right) entries are 1 and rest of the entries are zero. So, I 1 = [1], I 2 = [ ], I 3 =
7 Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Properties of the Identity Matrix Let A be a m n matrix. Then AI n = A I m A = a If A is a square matrix of size n n, then AI n = I n A = A. I n is called the Identity matrix of order n. Because of above, we say that I n is the multiplicative identity for the set of all square matrices of order n.
8 Proof. Preview Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Proof. We will prove for 3 3 matrices A. Write a b c A = u v w So, x y z AI 3 = a b c u v w x y z = Similarly, I 3 A = A. The proof is complete. a b c u v w x y z = A
9 Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Use of Matrix Algebra to solve systmes of linear equation Now that we know some Matrix Algebra, we will use this to give a proof of the following theorem that was stated before: Theorem. For a system of linear equations in n variables, precisely one of the following is true: The system has exactly one solution. The system has an infinite number of solutions. The system has no solution.
10 The Proof. Preview Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions Write the equation in the matrix form Ax = b. If the first or the last statements are true then there is nothing to prove. So, assume both are false. So, the system has at least two distint solutions x 1,x 2 with x 1 x 2. So, With y = x 1 x 2 0 we have Ax 1 = b and Ax 2 = b. Ay = A(x 1 x 2 ) = Ax 1 Ax 2 = b b = 0.
11 Zero Matrices Algebra of Matrix Multiplication Identity Matrix Number of Solutions For any scalar c, we have A(x 1 +cy) = Ax 1 +cay = b+0 = b So, x 1 +cy is a solution of the given system Ax = b, for all scalars c, which is infinitely many. The proof is complete.
12 Definition Algebra of Transpose Definition of Definition. Given a m n matrix A, the transpose of A, denoted by A T, is formed by writing the the columns of A as rows (equivalently, writing the rows as columns). So, transpose A T of A = is given by: a 11 a 12 a 13 a 1n a 21 a 22 a 13 a 2n a 31 a 32 a 33 a 3n a m1 a m2 a m3 a mn an m n matrix
13 Definition Algebra of Transpose : Continued A T = a 11 a 21 a 31 a m1 a 12 a 22 a 32 a m2 a 13 a 23 a 33 a m3 a 1n a 2n a 3n a mn an n m matrix Reading Assignment: Example 8 from 2.2 of the textbook.
14 Properties of Transpose Definition Algebra of Transpose Let A,B be matrices and c be a scalar. Assume all the multiplications below are defined. Then, (A T ) T = A Double transpose of A is itself (A+B) T = A T +B T transpose of sum (ca) T = ca T transpose of scalar multiplication (AB) T = B T A T transpose of product. Again, to prove we check entrywise equalities. Reading Assignment: Example 9 from 2.2 of the textbook.
15 Example: Noncommutativity Example: NonCancellation Let me draw your attention, how algebra of matrices differ from that of the algebra of real numbers: Matrix product is not commutative. That means AB BA, for some matrices A,B. See 2.2 Example 4. Cancellation property fails. That means there are matrices A,B,C, with C 0, such that See 2.2 Example 5. AC = BC but A B.
16 Example: Noncommutativity Example: NonCancellation Example of noncommutativity AB BA: We have [ ][ ] [ ] = [ ][ ] [ ] = Right hand sides of these two equations are not equal. So, commitativity fails for these two matrices.
17 Example: Noncommutativity Example: NonCancellation Example: AC = BC but A B: We have [ ][ ] = [ ] = [ So, cancellation property fails for matrix product. ][ ]
18 Solve for Matrix X Solve for the matrix X when 1 1 A = 3 1 B = (1) 5X +3A = 2B (2) 2B +4A = 5X (3) X +2A 2B = O
19 Solution Preview For (1) 5X +3A = 2B = X = 2 5 B 3 5 A = =
20 Solution: Continued For (2), X = 2 5 B A = = For (3) X = 2A+2B = =
21 Example Let A = C =,B = Demonstrate AC = BC bur A B ,
22 Solution: Preview We have AC = BC = So, AC = BC = = ,
23 Example Preview Let f(x) = x 2 2x +2 Compute f(a). A = Solution: We have A 2 = =
24 Solution: Continued Let f(a) = A 2 2A+2I 3 = =
25 Preview 2.2: Exwecise 14a, 14c, 15, 19, 23, 25, 31, 32, 33, 34, 55
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