Find the minors and cofactors of the elements | vedclass

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(A) The minors $(M_{ij})$ and cofactors $(A_{ij})$ are calculated as follows:$M_{11} = \left|\begin{array}{cc}0 & 4 \ 5 & -7\end{array}ight| = 0 -

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(A) The minors $(M_{ij})$ and cofactors $(A_{ij})$ are calculated as follows:$M_{11} = \left|\begin{array}{cc}0 & 4 \ 5 & -7\end{array}ight| = 0 - 20 = -20; \quad A_{11} = (-1)^{1+1}(-20) = -20$$M_{12} = \left|\begin{array}{cc}6 & 4 \ 1 & -7\end{array}ight| = -42 - 4 = -46; \quad A_{12} = (-1)^{1+2}(-46) = 46$$M_{13} = \left|\begin{array}{cc}6 & 0 \ 1 & 5\end{array}ight| = 30 - 0 = 30; \quad A_{13} = (-1)^{1+3}(30) = 30$$M_{21} = \left|\begin{array}{cc}-3 & 5 \ 5 & -7\end{array}ight| = 21 - 25 = -4; \quad A_{21} = (-1)^{2+1}(-4) = 4$$M_{22} = \left|\begin{array}{cc}2 & 5 \ 1 & -7\end{array}ight| = -14 - 5 = -19; \quad A_{22} = (-1)^{2+2}(-19) = -19$$M_{23} = \left|\begin{array}{cc}2 & -3 \ 1 & 5\end{array}ight| = 10 + 3 = 13; \quad A_{23} = (-1)^{2+3}(13) = -13$$M_{31} = \left|\begin{array}{cc}-3 & 5 \ 0 & 4\end{array}ight| = -12 - 0 = -12; \quad A_{31} = (-1)^{3+1}(-12) = -12$$M_{32} = \left|\begin{array}{cc}2 & 5 \ 6 & 4\end{array}ight| = 8 - 30 = -22; \quad A_{32} = (-1)^{3+2}(-22) = 22$$M_{33} = \left|\begin{array}{cc}2 & -3 \ 6 & 0\end{array}ight| = 0 + 18 = 18; \quad A_{33} = (-1)^{3+3}(18) = 18$Given $a_{11}=2, a_{12}=-3, a_{13}=5$ and $A_{31}=-12, A_{32}=22, A_{33}=18$,we verify:$a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33} = 2(-12) + (-3)(22) + 5(18) = -24 - 66 + 90 = -90 + 90 = 0$.

Find the minors and cofactors of the elements of the determinant $\left|\begin{array}{ccc}2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7\end{array}\right|$ and verify that $a_{11} A_{31}+a_{12} A_{32}+a_{13} A_{33}=0$.

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