Find the inverse of each of the matrices, if it exists.
Question:

Find the inverse of each of the matrices, if it exists.

$\left[\begin{array}{rr}3 & 10 \\ 2 & 7\end{array}\right]$

 

Solution:

Let $A=\left[\begin{array}{rr}3 & 10 \\ 2 & 7\end{array}\right]$

We know that A = IA

$\therefore\left[\begin{array}{rr}3 & 10 \\ 2 & 7\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] A$

$\Rightarrow\left[\begin{array}{ll}1 & 3 \\ 2 & 7\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\mathrm{R}_{2}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 3 \\ 0 & 1\end{array}\right]=\left[\begin{array}{lr}1 & -1 \\ -2 & 3\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-2 \mathrm{R}_{1}\right)$

$\Rightarrow\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{lr}7 & -10 \\ -2 & 3\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-3 \mathrm{R}_{2}\right)$

$\therefore A^{-1}=\left[\begin{array}{lr}7 & -10 \\ -2 & 3\end{array}\right]$

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