Chapter 16.3, Problem 9OE

To determine

To find:the dimensions of the given matrices and decide whether the product of the matrices is possible.

Answer to Problem 9OE

Dimension of first matrix $\left[\begin{array}{cc}\frac{2}{3}& 0\\ \frac{1}{5}& \frac{1}{2}\end{array}\right]$ is $2\times 2$ .

Dimension of second matrix $\left[\begin{array}{cc}\frac{1}{6}& \frac{1}{10}\\ \frac{1}{3}& 0\end{array}\right]$ is $2\times 2$ .

The product of given matrices ispossible.

Explanation of Solution

Given:

The matrices:

  $\left[\begin{array}{cc}\frac{2}{3}& 0\\ \frac{1}{5}& \frac{1}{2}\end{array}\right],\text{and}\left[\begin{array}{cc}\frac{1}{6}& \frac{1}{10}\\ \frac{1}{3}& 0\end{array}\right]$

Concept Used:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $B=\left[b_{ij}\right]$ is an $n\times p$ matrix, then the product AB is an $m\times p$ matrix given by $AB=\left[c_{ij}\right]$ , where $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

Calculation:

Dimension of a matrix with m rows and n columns can be written as

  $m\times n$

Consider the given matrices.

  $\left[\begin{array}{cc}\frac{2}{3}& 0\\ \frac{1}{5}& \frac{1}{2}\end{array}\right],\text{and}\left[\begin{array}{cc}\frac{1}{6}& \frac{1}{10}\\ \frac{1}{3}& 0\end{array}\right]$

Number of rows of the first matrix is 2 and number of columns is2.

Dimension of first matrix is $2\times 2$ .

Number of rows of the second matrix is 2 and number of columns is 2.

Dimension of second matrix is $2\times 2$ .

Note that here number of columns of first matrixis 2and the number of rows of 2^nd matrix is2.

Thus, product of given matrices is possible.