Question

For the determinant 2 -3 5 .
6 0 4
1 5 -7
Find M₁₂ and C₂₃ where M₁₂ is a minor of the element in first row and second column & C₂₃ is a cofactor of the element in the second row and third column.

Answer 1

$$\begin{gathered} \mathrm{D}=\left|\begin{array}{ccc}2& -3& 5\\ 6& 0& 4\\ 1& 5& -7\end{array}\right| \\ \mathrm{To}\mathrm{get}{\mathrm{M}}_{\mathrm{ij}}\mathrm{we}\mathrm{hide}{\mathrm{i}}^{\mathrm{th}}\mathrm{row}\mathrm{and}{\mathrm{j}}^{\mathrm{th}}\mathrm{column}.\mathrm{So}\mathrm{for}{\mathrm{M}}_{12}: \\ {\mathrm{M}}_{12}=\left|\begin{array}{cc}6& 4\\ 1& -7\end{array}\right|=6(-7)-\left(4\right)\left(1\right)=-46 \\ {\mathrm{C}}_{\mathrm{ij}}=(-1{)}^{\mathrm{i}+\mathrm{j}}{\mathrm{M}}_{\mathrm{ij}} \\ \Rightarrow {\mathrm{C}}_{23}=(-1{)}^5{\mathrm{M}}_{23}=(-1)\left|\begin{array}{cc}2& -3\\ 1& 5\end{array}\right|=(-1)\left[\right(5\left)\right(2)-(-3\left)\right(1\left)\right]=-13 \end{gathered}$$