Question

What is the inverse of the matrix $\left[\begin{array}{cc}-3& 2\\ 5& -1\end{array}\right]$

• A. $\left[\begin{array}{cc}3& 5\\ 2& 1\end{array}\right]$
• B. $\frac{1}{7}\left[\begin{array}{cc}3& 5\\ 2& 1\end{array}\right]$
• C. $\frac{1}{7}\left[\begin{array}{cc}-1& -2\\ -5& -3\end{array}\right]$
• D. $\frac{1}{7}\left[\begin{array}{cc}1& 2\\ 5& 3\end{array}\right]$

Answer 1

The correct option is D $\frac{1}{7}\left[\begin{array}{cc}1& 2\\ 5& 3\end{array}\right]$

It is always possible to find the inverse of a matrix by cofactor method. But for $2\times 2$ matrix there is a short cut method.
So for a $2\times 2$ matrix of the form$\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, to find the inverse.
1) Swap the positions of a and d.
2) take negatives of b and c
3) divide everything by determinant of the matrix.
$\therefore$ Inverse of the given matrix.
${\left[\begin{array}{cc}-3& 2\\ 5& -1\end{array}\right]}^{-1}=\frac{1}{-7}\left[\begin{array}{cc}-1& -2\\ -5& -3\end{array}\right]=\frac{1}{7}\left[\begin{array}{cc}1& 2\\ 5& 3\end{array}\right]$