Q. 11 - Precalculus Enhanced with Graphing Utilities

Question

find the inverse of each nonsingular matrix.

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

Step-by-Step Solution

Answer

The inverse of the given matrix is

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

Step 1. Given

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

Step 2. Calculation of A | I 2

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 $\left[A|I_2\right]$

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

Step 4. Final Calculation

Perform

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

$$\begin{gathered} Thus we see the inverse as \\ \left[\begin{array}{cc}2& -1\\ -\frac{5}{2}& \frac{3}{2}\end{array}\right] \end{gathered}$$