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Show that the adjoint of $A=\begin{bmatrix} -1 & -2 & -2 \\2 & 1 & -2 \\2 & -2 & 1 \end{bmatrix}$ is$\; 3A^T.$

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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} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}$
The cofactor matrix $ [ A_{ij}] = \begin{bmatrix} (1-4) & -(2+4) & (-4-2) \\ -(2-4) & (-1+4) & -(2+4) \\ (4+2) & -(2+4) & (-1+4) \end{bmatrix}$
$ = \begin{bmatrix} -3 & -6 & -6 \\ 6 & 3 & -6 \\ 6 & -6 & 3 \end{bmatrix}= 3\begin{bmatrix} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix}$
$ adj\: A = [A_{ij}]^T = 3\begin{bmatrix} -1 & 2 & 2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{bmatrix}$
Step 2
$ 3\: A^T = 3\begin{bmatrix} -1 & -2 & -2 \\ 2 & 1 & -2 \\ 2 & -2 & 1 \end{bmatrix} = 3\begin{bmatrix} -1 & 2 & 2 \\ -2 & 1 & -2 \\ -2 & -2 & 1 \end{bmatrix}$
From step 1 $ adj\: A = 3A^T$
answered May 23, 2013 by thanvigandhi_1
 

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