The adjoint of the matrix [ ll 1 & 2 3 & 4 ] is - Vidyadip

Question

The adjoint of the matrix $\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ is

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Answer

Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ then $| A |=\left|\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right|$.
Now, cofactors of elements of $| A |$ are
$C_{11}=(-1)^{1+1} (4)=4$
$C_{12}=(-1)^{1+2}(3)=-3$
$C_{21}=(-1)^{2+1}(2)=-2$
and $C_{22}=(-1)^{2+2}(1)=1$
Now, adj $(A) =\left[\begin{array}{ll}C_{11} & C_{12} \\ C_{21} & C_{22}\end{array}\right]^T$
$\begin{array}{l}=\left[\begin{array}{rr}4 & -3 \\ -2 & 1\end{array}\right]^T \end{array} $
$ =\left[\begin{array}{rr}4 & -2 \\ -3 & 1\end{array}\right]$

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