Question

On expanding by first row, the value of the determinant of 3 × 3 square matrix $A=\left[a_{ij}\right]\mathrm{is}{a}_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$, where [C_ij] is the cofactor of a_ij in A. Write the expression for its value on expanding by second column.

Answer 1

If $A=\left[{\mathrm{a}}_{\mathrm{i}\mathrm{j}}\right]$ is a square matrix of order n, then the sum of the products of elements of a row (or a column) with their cofactors is always equal to det (A). Therefore,

$$\begin{gathered} \sum _{i=1}^n{a}_{ij}{C}_{ij}=\left|A\right|\mathrm{and}\sum _{j=1}^n{a}_{ij}{C}_{ij}=\left|A\right| \\ \mathrm{Given}:\left|A\right|={\mathrm{a}}_{11}{\mathrm{C}}_{11}+{\mathrm{a}}_{12}{\mathrm{C}}_{12}+{\mathrm{a}}_{13}{\mathrm{C}}_{13}\left[\mathrm{Expanding}\mathrm{along}{R}_1\right] \\ \mathrm{Now}, \\ \left|A\right|=a_{12}{\mathrm{C}}_{12}+{\mathrm{a}}_{22}{\mathrm{C}}_{22}+{\mathrm{a}}_{32}{\mathrm{C}}_{32}\left[\mathrm{Expanding}\mathrm{along}{R}_2\right]\left[{\mathrm{a}}_{12},{\mathrm{a}}_{22}\mathrm{and}{\mathrm{a}}_{32}\mathrm{are}\mathrm{elements}\mathrm{of}{C}_2\right] \end{gathered}$$