Chapter 11.4, Problem 91AYU

To determine

To explain: When two matrices A and B are multiplied together to form AB then number columns in matrix A should be equal to number of rows of matrix B.

Answer to Problem 91AYU

The product of two matrices is computed when each row of first matrix is multiplied with each column of second matrix.

Explanation of Solution

Given information:

Two matrices A and B.

The statement is when two matrices A and B are multiplied together to form AB then number columns in matrix A should be equal to number of rows of matrix B.

Number of columns in matrix A should be equal to number of rows of matrix B because first row of first matrix is multiplied with first column of second matrix. Number of elements in a row depend on number of column that matrix has so, number columns in matrix A should be equal to number of rows of matrix B.

For example:

Consider the matrices are $\left[\begin{array}{ccc}4& -2& 3\\ 0& 1& 2\\ -1& 0& 1\end{array}\right]\left[\begin{array}{cc}2& 6\\ 1& -1\\ 0& 0\end{array}\right]$

Recall that to find the product AB of two matrices A and B, the number of column in matrix A must be equal to number of rows in matrix B.

  $\begin{array}{c}\left[\begin{array}{ccc}4& -2& 3\\ 0& 1& 2\\ -1& 0& 1\end{array}\right]\left[\begin{array}{cc}2& 6\\ 1& -1\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}8-2+0& 24+2+0\\ 0+1+0& 0-1+0\\ -2+0+0& -6+0+0\end{array}\right]\\ =\left[\begin{array}{cc}6& 26\\ 1& -1\\ -2& -6\end{array}\right]\end{array}$

Thus, product of two matrices is computed when each row of first matrix is multiplied with each column of second matrix.