Question

find the inverse of each nonsingular matrix.

$\left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]$

Short answer

The inverse of the given matrix is

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

Step-by-step solution

Step 1. Given

$A=\left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]$

Step 2. Calculation of A|I2

Given A as $2\times 2$matrix therefore $\left[A|I_2\right]$where $I_2$is identity matrix of second order.

$\left[A|I_2\right]=\left[\begin{array}{cccc}3& 2& 1& 0\\ 5& 4& 0& 1\end{array}\right]$

Step 3. Calculation for transformation

Transforming matrix $\left[A|I_2\right]$

$$\begin{gathered} performtherowoperation \\ {R}_1=\frac{1}{3}{r}_1\&R_2=r_2-5r_1 \\ \left[\begin{array}{cccc}3& 2& 1& 0\\ 5& 4& 0& 1\end{array}\right]\to \left[\begin{array}{cccc}1& \frac{2}{3}& \frac{1}{3}& 0\\ 5& 4& 0& 1\end{array}\right] \\ \to \left[\begin{array}{cccc}1& \frac{2}{3}& \frac{1}{3}& 0\\ 0& \frac{2}{3}& -\frac{5}{3}& 1\end{array}\right] \end{gathered}$$

Step 4. Final Calculation

Perform

$$\begin{gathered} R_1=r_1-r_2\&R_2=\frac{3}{2}{r}_2 \\ \left[\begin{array}{cccc}1& \frac{2}{3}& \frac{1}{3}& 0\\ 0& \frac{2}{3}& -\frac{5}{3}& 1\end{array}\right]\to \left[\begin{array}{cccc}1& 0& 2& -1\\ 0& \frac{2}{3}& -\frac{5}{3}& 1\end{array}\right] \\ \to \left[\begin{array}{cccc}1& 0& 2& -1\\ 0& 1& -\frac{5}{2}& \frac{3}{2}\end{array}\right] \end{gathered}$$

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