Question

In question 18, write the value of a₁₁ C₂₁ + a₁₂ C₂₂ + a₁₃ C₂₃.

Answer 1

We know that in a square matrix of order n, the sum of the products of elements of a row (or a column) with the cofactors of the corresponding elements of some other row (or column ) is zero. Therefore,

$$\begin{gathered} A=\left[a_{ij}\right]\mathrm{is}\mathrm{a}\mathrm{square}\mathrm{matrix}\mathrm{of}\mathrm{order}\mathit{}n\mathit{.} \\ \underset{j=1}{\overset{n}{\Rightarrow \sum }}{a}_{ij}{C}_{kj}=0\mathrm{and}\sum _{i=1}^n{a}_{ij}{C}_{ik}=0 \\ \Rightarrow a_{11}{C}_{21}+a_{12}{C}_{22}+a_{13}{C}_{23}=0\left[\mathrm{Since}\mathrm{the}\mathrm{elements}\mathrm{are}\mathrm{of}\mathrm{first}\mathrm{row}\mathrm{and}\mathrm{the}\mathrm{cofactors}\mathrm{are}\mathrm{of}\mathrm{elements}\mathrm{of}\mathrm{second}\mathrm{row}\right] \\ \Rightarrow {\mathrm{a}}_{11}{\mathrm{C}}_{21}+{\mathrm{a}}_{12}{\mathrm{C}}_{22}+{\mathrm{a}}_{13}{\mathrm{C}}_{23}=0 \end{gathered}$$