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# Find the adjoint of the following matrices $\begin{bmatrix} 3 & -1 \\2 & -4& \end{bmatrix}$

• Let A = $[ a_{ij} ]$ be a square matrix of order x. Let $A_{ij}$ be the cofactor of $a _{ij}$. Then $[ A_{ij} ]$ is the matrix of cofactors and $adj\: A$ ( or adjoint of the matrix A ) is given by $adj\: A = [ A_{ij} ]^T$
• Let A be a square matrix of order 2. $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then $adj A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ ( shorcut for finding the adjoint )
Let A =$\begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix}$
$[ A_{ij} ]= \begin{bmatrix} -4 & -2 \\ 1 & 3 \end{bmatrix}$
$adjA = [ A_{ij} ]^T = \begin{bmatrix} -4 & 1 \\ -2 & 3 \end{bmatrix}$
or by shortcut method: If $A = \begin{bmatrix} 3 & -1 \\ 2 & -4 \end{bmatrix}$ $\therefore adjA = \begin{bmatrix} -4 & 1 \\ -2 & 3 \end{bmatrix}$