Question

Find adjoint of matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$

We know that $adjA=C^T$

So, let's find cofactors of each of the elements of A.

$C_{11}=(-1{)}^{1+1}4=4$

$C_{12}=(-1{)}^{1+2}3=-3$

$C_{21}=(-1{)}^{2+1}2=-2$

$C_{22}=(-1{)}^{2+2}1=1$

So, cofactor matrix is $C=\left[\begin{array}{cc}4& -3\\ -2& 1\end{array}\right]$

$\Rightarrow C^T=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$

Hence, $adjA=\left[\begin{array}{cc}4& -2\\ -3& 1\end{array}\right]$