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Q)

Find the adjoint of the following matrics;$\begin{bmatrix} 2& 5 & 3 \\3 & 1 & 2 \\1 & 2 & 1 \end{bmatrix}$

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A)
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  • 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
$ A = \begin{bmatrix} 2 & 5 & 3 \\ 3 & 1 & 2 \\ 1 & 2 & 1 \end{bmatrix} $
$ [A_{ij}] = \begin{bmatrix} (1-4) & -(3-2) & (6-1) \\ -(5-6) & (2-3) & -(4-5) \\ (10-3) & -(4-9) & (2-15) \end{bmatrix} = \begin{bmatrix} -3 & -1 & 5 \\ 1 & -1 & 1 \\ 7 & 5 & -13 \end{bmatrix} $
Step 2
$ adj\: A = [ A_{ij}]^T = \begin{bmatrix} -3 & 1 & 7 \\ -1 & -1 & 5 \\ 5 & 1 & -13 \end{bmatrix} $
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