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# Find the adjoint of the matrix: $\begin{bmatrix} 1&2 \\ 3&4 \end{bmatrix}$

$\begin{array}{1 1} \begin{bmatrix}4 & -2\\-3& 1\end{bmatrix} \\ \begin{bmatrix}4 & 2\\-3& 1\end{bmatrix} \\ \begin{bmatrix}4 & -2\\3& 1\end{bmatrix} \\ \begin{bmatrix}4 & -2\\-3& -1\end{bmatrix}\end{array}$

Toolbox:
• (i)A square matrix is said to be singular if |A|$\neq$ 0.
• (ii)The adjoint of a square matrix is defined as the transpose of the matrix $[A_{ij}]_{n\times n}$.where $A_{ij}$ is the cofactor of the element $a_{ij}.$Adjoint of the matrix is denoted by adj A.
We know $A_{ij}=(-1)^{i+j}M_{ij}$
Here $M_{11}=4$
$M_{12}=3$
$M_{21}=2$
$M_{22}=1$
$A_{11}=(-1)^{1+1}\times 4=4$
$A_{12}=(-1)^{1+2}\times 3=-3$
$A_{21}=(-1)^{2+1}\times 2=-2$
$A_{11}=(-1)^{2+2}\times 1=1$
Therefore adj A=$\begin{bmatrix}A_{11} & A_{21}\\A_{12} & A_{22}\end{bmatrix}=\begin{bmatrix}4 & -2\\-3& 1\end{bmatrix}$
Therefore adj A=$\begin{bmatrix}4 & -2\\-3& 1\end{bmatrix}$