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(C) The sum of the products of elements of any row (or column) of a determinant with their corresponding cofactors is equal to the value of the determ

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(C) The sum of the products of elements of any row (or column) of a determinant with their corresponding cofactors is equal to the value of the determinant. However,the sum of the products of elements of one row with the cofactors of another row is always zero.Specifically,for a matrix $A$,the property states that $a_{i1}A_{j1} + a_{i2}A_{j2} + a_{i3}A_{j3} = 0$ for $i 
eq j$.Here,we are asked to find $A_{31} + A_{32} + A_{33}$.Consider the determinant of matrix $A$ expanded along the first row:$|A| = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$.Now,consider the expression $a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33}$. Since this is the sum of products of elements of the first row with the cofactors of the third row,this sum must be equal to $0$.Substituting the values of $a_{11} = 1$,$a_{12} = 1$,and $a_{13} = 1$ from the matrix $A$:$1 \cdot A_{31} + 1 \cdot A_{32} + 1 \cdot A_{33} = 0$Therefore,$A_{31} + A_{32} + A_{33} = 0$.

If $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & -3 \\ -1 & 2 & 3 \end{bmatrix}$,then find the value of $A_{31} + A_{32} + A_{33}$,where $A_{ij}$ denotes the cofactor of the element $a_{ij}$ of matrix $A$.

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