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# Find the adjoint of the following matrices:$\begin{bmatrix} 1 & 2 & 3 \\0 & 5 & 0 \\2 & 4 & 3 \end{bmatrix}$

Toolbox:
• Let $A = [ a_{ij} ]$ be a square matrix 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 $adj\: A=[A_{ij}]^T$
Step 1
Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 0 \\ 2 & 4 & 3 \end{bmatrix}$
$[ A_{ij} ] = \begin{bmatrix} (15-0) & -(0-0) & (0-10) \\ -(6-12) & (3-6) & -(4-4) \\ (0-15) & -(0-0) & (5-0) \end{bmatrix} = \begin{bmatrix} 15 & 0 & -10 \\ 6 & -3 & 0 \\ -15 & 0 & 5 \end{bmatrix}$
Step 2
$adj\: A = [ A_{ij} ]^T$
$= \begin{bmatrix} 15 & 6 & -15 \\ 0 & -3 & 0 \\ -10 & 0 & 5 \end{bmatrix}$

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