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 matrixB .

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

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