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  4. The inverse of the matrix A=[2 1 7 4] is

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Q. The inverse of the matrix $A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$ is

Matrices

Solution:

Let $A=\begin{bmatrix}2 & 1 \\ 7 & 4\end{bmatrix}$. We know that, $A=I A$
$\therefore \begin{bmatrix}2 & 1 \\7 & 4\end{bmatrix}=\begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix} A \Rightarrow\begin{bmatrix}1 & 1 / 2 \\7 & 4\end{bmatrix}=\begin{bmatrix}1 / 2 & 0 \\0 & 1\end{bmatrix} A$ $\left(\right.$ using $\left.R_1 \rightarrow \frac{1}{2} R_1\right)$
$\Rightarrow \begin{bmatrix}1 & 1 / 2 \\ 0 & 1 / 2\end{bmatrix}=\begin{bmatrix}1 / 2 & 0 \\ -7 / 2 & 1\end{bmatrix} A $ (using $\left.R_2 \rightarrow R_2-7 R_1\right)$
$\Rightarrow \begin{bmatrix}1 & 1 / 2 \\ 0 & 1\end{bmatrix}=\begin{bmatrix}1 / 2 & 0 \\ -7 & 2\end{bmatrix} A $ (using $R_2 \rightarrow 2 R_2$ )
$\Rightarrow \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}=\begin{bmatrix}4 & -1 \\ -7 & 2\end{bmatrix}A \left(\right.$ using $\left.R_1 \rightarrow R_1-\frac{1}{2} R_2\right)$
$\therefore A^{-1}=\begin{bmatrix}4 & -1 \\ -7 & 2\end{bmatrix}$