To determine The value of expressions AB , AB T , BA , B T A . Explanation Given: The matrices are A = [ 4 − 2 3 − 2 1 6 1 2 2 ] , B = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] , C = [ 0 1 3 2 − 2 0 ] , a = [ 1 − 2 0 ] , b = [ 3 1 − 1 ] . Definition used: “Multiplication of matrices: The product C = AB (in this order) of an m × n matrix A = [ a j k ] times an r × p matrix B = [ b j k ] is defined if and only if r = n and is then the m × p matrix C = [ c j k ] with entries c j k = ∑ l = 1 n a j l b l k = a j 1 b 1 k + a j 2 b 2 k + ⋯ + a j n b n k , where j = 1 , 2 , … , m and k = 1 , 2 , … , p .” Transpose of a matrix: “The transpose of an m × n matrix A = a j k is the n × m matrix A T that has the first row of A as its first column , the second row of A as its second column , and so on. Thus the transpose of A is A T = [ a k j ] .” Calculation: Use the above mentioned definition, find the value of AB as follows. AB = [ 4 − 2 3 − 2 1 6 1 2 2 ] [ 1 − 3 0 − 3 1 0 0 0 − 2 ] = [ 4 + 6 + 0 − 12 − 2 + 0 0 + 0 − 6 − 2 − 3 + 0 6 + 1 + 0 0 + 0 − 12 1 − 6 + 0 − 3 + 2 + 0 0 + 0 − 4 ] = [ 10 − 14 − 6 − 5 7 − 12 − 5 − 1 − 4 ] Use the definition transpose and find the transpose of B . B = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] B T = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] = B Since B T = B , the value of AB T = AB

Custom Kreyszig: Advanced Engineering Mathematics

10th Edition

ISBN: 9781119166856

Author: Kreyszig

Publisher: JOHN WILEY+SONS INC.CUSTOM

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Chapter 7.2, Problem 11P

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The value of expressions AB,ABT,BA,BTA.

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Custom Kreyszig: Advanced Engineering Mathematics

Ch. 7.1 - Equality. Give reasons why the five matrices in...Ch. 7.1 - Double subscript notation. If you write the matrix...Ch. 7.1 - Sizes. What sizes do the matrices in Examples 1,...Ch. 7.1 - Main diagonal. What is the main diagonal of A in...Ch. 7.1 - Scalar multiplication. If A in Example 2 shows the...Ch. 7.1 - If a 12 × 12 matrix A shows the distances between...Ch. 7.1 - Addition of vectors. Can you add: A row and a...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...Ch. 7.1 - Let Find the following expressions, indicating...

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