Question

What is the inverse of the matrix $A=\left[\begin{array}{cc}3& 1\\ 7& 9\end{array}\right]$

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Given that $A=\left|\begin{array}{cc}3& 1\\ 7& 9\end{array}\right|$

$|A|=3\times 9-1\times 7=20$

Cofactor matrix of $A=\left[\begin{array}{cc}9& -7\\ -1& 3\end{array}\right]$

Adj $\left(A\right)=\left[\begin{array}{cc}9& -1\\ -7& 3\end{array}\right]$

$A^{-1}=\frac{1}{|A|}.Adj\left(A\right)$

$=\frac{1}{20}.\left[\begin{array}{cc}9& -1\\ -7& 3\end{array}\right]$

Hence correct option (c)

Here we can see that finding adjoint can be found out using a simple method of interchanging $a_{11}$ and $a_{22}$ and reversing the signs of $a_{12}$ and $a_{21}$. This is applicable to all $2\times 2$ matrices.

Hence

${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^{-1}=\frac{a}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$