   Chapter 1.6, Problem 11TFE

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# Label each of the following statements as either true or false.Let A be in M m     ×     n   ( ℝ ) and B be in M n     ×     p ( ℝ ) . If the i th row of A contains all zeros, then the i th row of A B contains all zeros.

To determine

Whether the statement, “Let A be in Mm×n() and B be in Mn×p(). If i th row of A contains all zeros, then i th row of AB contains all zeros” is true or false.

Explanation

Consider the statement, “Let A be in Mm×n() and B be in Mn×p(). If i th row of A contains all zeros, then i th row of AB contains all zeros.”

By definition of matrix multiplication,

The product of m×n matrix A over and n×p matrix B over is m×p matrix C=AB, where the element cij in row i and column j of AB is found by using the elements in row i of A, and the elements in column j of B in the following manner:

columnjofBcolumnjofCrowiofA[ai1ai2ai3ain

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