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

MCQ

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

A
$\left[\begin{array}{cc}4 & -2 \\ -3 & -1\end{array}\right]$
✓
$\left[\begin{array}{cc}4 & -2 \\ -3 & 1\end{array}\right]$
C
$\left[\begin{array}{cc}4 & -3 \\ -2 & 1\end{array}\right]$
D
$\left[\begin{array}{ll}4 & -2 \\ 1 & -3\end{array}\right]$

✓

Answer

Correct option: B.

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

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, $\text{adj} (A) =\left[\begin{array}{ll}C_{11} & C_{12} \\ C_{21} & C_{22}\end{array}\right]^T$
$\begin{array}{l}=\left[\begin{array}{rr}4 & -3 \\ -2 & 1\end{array}\right]^T \\ =\left[\begin{array}{rr}4 & -2 \\ -3 & 1\end{array}\right]\end{array}$

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