Chapter 7.6, Problem 33E

To determine

To calculate: the inverse of the matrix- $\left[\begin{array}{cc}2& 3\\ -1& 5\end{array}\right]$

Answer to Problem 33E

Inverse of the given matrix is:

  $\left[\begin{array}{cc}\frac{5}{13}& -\frac{3}{13}\\ \frac{1}{13}& \frac{2}{13}\end{array}\right]$

Explanation of Solution

Given information:

The following matrix:

  $\left[\begin{array}{cc}2& 3\\ -1& 5\end{array}\right]$

Formula used:

Inverse of a $2\times 2$ matrix is given by,

  $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

when $ad-bc\ne 0$ .

Calculation:

For the given matrix, apply the formula for inverse of a $2\times 2$ matrix to obtain:

  $\begin{array}{l}ad-bc=(2)(5)-(3)(-1)\text{[Substitute]}\\ =10+3\text{[Multiply]}\\ =13\text{[Simplify]}\end{array}$

Because this quantity is not zero, the inverse is formed by interchanging the entries on the main diagonal, changing the signs of the other two entries, and multiplying by scalar $\frac{1}{13}$ as follows,

  $\begin{array}{l}{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right][\text{Formula]}\\ =\frac{1}{13}\left[\begin{array}{cc}5& -3\\ 1& 2\end{array}\right]\text{[Substitute]}\\ \text{=}\left[\begin{array}{cc}\frac{5}{13}& -\frac{3}{13}\\ \frac{1}{13}& \frac{2}{13}\end{array}\right]\left[\text{Multiply by scalar }\frac{1}{13}\right]\end{array}$

Hence, inverse of the given matrix is

  $\left[\begin{array}{cc}\frac{5}{13}& -\frac{3}{13}\\ \frac{1}{13}& \frac{2}{13}\end{array}\right]$