If matrix A = begin{bmatrix} 1 & 2 & 3 1 & 1 | vedclass

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(C) We know that the sum of the product of elements of any row (or column) with their corresponding cofactors is equal to the determinant of the matri

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$-1$

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(C) We know that the sum of the product of elements of any row (or column) with their corresponding cofactors is equal to the determinant of the matrix,i.e.,$a_{i1} A_{i1} + a_{i2} A_{i2} + a_{i3} A_{i3} = |A|$.Here,$a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33} = |A|$.Now,we calculate the determinant of $A$ by expanding along the first row:$|A| = 1(1 	imes 7 - 5 	imes 4) - 2(1 	imes 7 - 5 	imes 2) + 3(1 	imes 4 - 1 	imes 2)$$|A| = 1(7 - 20) - 2(7 - 10) + 3(4 - 2)$$|A| = 1(-13) - 2(-3) + 3(2)$$|A| = -13 + 6 + 6$$|A| = -1$Therefore,the value is $-1$.

If matrix $A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 1 & 5 \\ 2 & 4 & 7 \end{bmatrix}$,then the value of $a_{31} A_{31} + a_{32} A_{32} + a_{33} A_{33}$ is

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