Find the adjoint of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \)
Answer
This gives \( Adj(A) = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{bmatrix} \)
Step 1 ：Step 1: Find the cofactor matrix of A. The cofactor of a matrix A is given by \(C_{ij} = (-1)^{i+j}det(M_{ij})\) where \(M_{ij}\) is the (i,j)th minor matrix.
Step 2 ：For A, we get \( C = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{bmatrix} \)
Step 3 ：Step 2: The adjoint of a matrix A is the transpose of its cofactor matrix. So the adjoint of A is \(Adj(A) = C^T\)
Step 4 ：So, we get \( Adj(A) = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{bmatrix}^T \)
Step 5 ：This gives \( Adj(A) = \begin{bmatrix} -3 & 6 & -3 \\ 6 & -12 & 6 \\ -3 & 6 & -3 \end{bmatrix} \)
