If begin{bmatrix} 5 & a & -7 b & -7 & c -7 | vedclass

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(A) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 3 & 1 \ 3 & 1 & 2 \end{bmatrix}$. The adjoint of a matrix is the transpose of its cofactor matrix.First

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(A) Let $A = \begin{bmatrix} 1 & 2 & 3 \ 2 & 3 & 1 \ 3 & 1 & 2 \end{bmatrix}$. The adjoint of a matrix is the transpose of its cofactor matrix.First,we calculate the cofactors $C_{ij}$ of matrix $A$:$C_{11} = (-1)^{1+1}(3 	imes 2 - 1 	imes 1) = 5$$C_{12} = (-1)^{1+2}(2 	imes 2 - 3 	imes 1) = -1$$C_{13} = (-1)^{1+3}(2 	imes 1 - 3 	imes 3) = -7$$C_{21} = (-1)^{2+1}(2 	imes 2 - 3 	imes 1) = -1$$C_{22} = (-1)^{2+2}(1 	imes 2 - 3 	imes 3) = -7$$C_{23} = (-1)^{2+3}(1 	imes 1 - 3 	imes 2) = 5$$C_{31} = (-1)^{3+1}(2 	imes 1 - 3 	imes 3) = -7$$C_{32} = (-1)^{3+2}(1 	imes 1 - 2 	imes 3) = 5$$C_{33} = (-1)^{3+3}(1 	imes 3 - 2 	imes 2) = -1$The cofactor matrix is $\begin{bmatrix} 5 & -1 & -7 \ -1 & -7 & 5 \ -7 & 5 & -1 \end{bmatrix}$.The adjoint is the transpose of this matrix: $	ext{adj}(A) = \begin{bmatrix} 5 & -1 & -7 \ -1 & -7 & 5 \ -7 & 5 & -1 \end{bmatrix}$.Comparing this with $\begin{bmatrix} 5 & a & -7 \ b & -7 & c \ -7 & d & -1 \end{bmatrix}$,we get $a = -1, b = -1, c = 5, d = 5$.Therefore,$a+b+c+d = -1 - 1 + 5 + 5 = 8$.

If $\begin{bmatrix} 5 & a & -7 \\ b & -7 & c \\ -7 & d & -1 \end{bmatrix}$ is the adjoint of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2 \end{bmatrix}$,then $a+b+c+d=$

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