Explanation Consider that both AB and BA are defined where neither A nor B is a square matrix. Recall that for matrices A with dimension m × n and B with dimension k × l , the product AB is defined if the number of columns in A equal number of columns in B , n = k . Let, A is m × n and B is n × m matrix. Since, number of columns in A , n , is equal to the number of rows in B , the product AB is defined. Since, number of columns in B , m , is equal to the number of rows in A , the product BA is defined. Thus, when A is m × n and B is n × m matrix and m ≠ n then both AB and BA are defined where neither A nor B is a square matrix. For example, consider the matrices A = [ 1 2 ] and B = [ 2 1 ] . Recall that for matrices A with dimension m × n and B with dimension k × l , the product AB is defined if the number of columns in A equal number of columns in B , n = k . Since, number of rows in A = [ 1 2 ] is 2 and number of columns is 1 , substitute 2 for m and 1 for n in m × n . Thus, dimension of A = [ 1 2 ] is 2 × 1 . Since, number of rows in B = [ 2 1 ] is 1 and number of columns is 2 , substitute 1 for m and 2 for n in m × n . Thus, dimension of B = [ 2 1 ] is 1 × 2 . Since, number of columns in A is equal to the number of rows in B , the product AB is defined. Recall that for matrices A with dimension m × n and B with dimension n × k , the product AB is the matrix of dimension m × k and ij th entry of AB is the sum of product of corresponding entries of row i of A and column j of B

7th Edition

ISBN: 9781337280426

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 4.2, Problem 82E

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The condition that both AB and BA are defined where neither A nor B is a square matrix.

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Chapter 4.2, Problem 81E

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The condition that both AB and BA are defined where neither A nor B is a square matrix.

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