Question

Find the inverse of matrices A,B,C and in Example 7.

Short answer

The inverse of matrix A, B, C are $\left[\begin{array}{cc}5& -2\\ -7& 3\end{array}\right],\frac{1}{26}\left[\begin{array}{cc}5& -3\\ 2& 4\end{array}\right],\frac{1}{2}\left[\begin{array}{cc}6& 0\\ -5& \frac{1}{3}\end{array}\right]$respectively.

Step-by-step solution

Formula to find Inverse

For a $2\times 2$matrix of the form$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the inverse is given by the formula:$A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Find the inverse of matrix A

The given matrix is $A=\left[\begin{array}{cc}3& 2\\ 7& 5\end{array}\right]$.

By using the inverse formula, find the inverse of matrix A

role="math" localid="1647469518122" $$\begin{gathered} A^{-1}=\frac{1}{3·5-2·7}\left[\begin{array}{cc}5& -2\\ -7& 3\end{array}\right] \\ =\frac{1}{15-14}\left[\begin{array}{cc}5& -2\\ -7& 3\end{array}\right] \\ =\left[\begin{array}{cc}5& -2\\ -7& 3\end{array}\right] \end{gathered}$$

Hence, the inverse of matrix is $\left[\begin{array}{cc}5& -2\\ -7& 3\end{array}\right]$.

Find the inverse of matrix B

The given matrix is$B=\left[\begin{array}{cc}4& 3\\ -2& 5\end{array}\right]$.

By using the inverse formula, find the inverse of matrix B

$$\begin{gathered} B^{-1}=\frac{1}{4·5-3·\left(-2\right)}\left[\begin{array}{cc}5& -3\\ 2& 4\end{array}\right] \\ =\frac{1}{20+6}\left[\begin{array}{cc}5& -3\\ 2& 4\end{array}\right] \\ =\frac{1}{26}\left[\begin{array}{cc}5& -3\\ 2& 4\end{array}\right] \end{gathered}$$

Hence, the inverse of matrix is $\frac{1}{26}\left[\begin{array}{cc}5& -3\\ 2& 4\end{array}\right]$.

Find the inverse of matrix C

The given matrix is$C=\left[\begin{array}{cc}\frac{1}{3}& 0\\ 5& 6\end{array}\right]$.

By using the inverse formula, find the inverse of matrix C

$$\begin{gathered} C^{-1}=\frac{1}{6·{\displaystyle \frac{1}{3}}-0·\left(-5\right)}\left[\begin{array}{cc}6& 0\\ -5& \frac{1}{3}\end{array}\right] \\ =\frac{1}{2}\left[\begin{array}{cc}6& 0\\ -5& \frac{1}{3}\end{array}\right] \end{gathered}$$

Hence, the inverse of matrix is$=\frac{1}{2}\left[\begin{array}{cc}6& 0\\ -5& \frac{1}{3}\end{array}\right]$