Question

On expanding by first row, the value of the determinant of 3 × 3 square matrix
  \[A = \left[ a_{ij} \right]\text{ is }a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13}\] , 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 \[A = \left[ a_{i j} \right]\]  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, 
\[\sum^n_{i = 1} a_{i j} C_{i j} = \left| A \right| and \sum^n_{j = 1} a_{i j} C_{i j} = \left| A \right|\] 
\[Given: \left| A \right| = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13} \left[\text{ Expanding along }R_1 \right]\] 
Now, 
\[\left| A \right| = a_{12} C_{12} + a_{22} C_{22} + a_{32} C_{32} \left[\text{ Expanding along }R_2 \right] \left[ a_{12} , a_{22}\text{ and }a_{32}\text{ are elements of }C_2 \right] \]