   Chapter 1.6, Problem 9TFE

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# Label each of the following statements as either true of false.The products A B and B A are defined if and only if both A and B are square matrices of the same order.

To determine

Whether the statement, “The product AB and BA are defined if and only if A and B are square matrices of same order” is true or false.

Explanation

Consider the statement, “The product AB and BA are defined if and only if A and B are square matrices of same order”.

Let A=2×3 and B=3×2

Therefore, by definition of matrix multiplication,

As the number of columns in A is equal to the number of rows in B, then the product AB is defined.

Therefore, AB has dimension 2×2.

AB=

By using definition of matrix multiplication,

=[(2)(1)+(0)(5)+(3)(1)(2)(2)+(0)(6)+(3)(1)(4)(1)+(1)(5)+(1)(1)(4)(2)+(1)(6)+(1)(1)]

=[2+034+0+34+518+6+1]

=2×2

Similarly, as the number of columns in B is equal to the number of rows in A, then the product BA is defined

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