Question

If $A=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$ and C_ij is cofactor of a_ij in A, then value of |A| is given by

(a) a₁₁ C₃₁ + a₁₂ C₃₂ + a₁₃ C₃₃
(b) a₁₁ C₁₁ + a₁₂ C₂₁ + a₁₃ C₃₁
(c) a₂₁ C₁₁ + a₂₂ C₁₂ + a₂₃ C₁₃
(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

Answer 1

(d) a₁₁ C₁₁ + a₂₁ C₂₁ + a₁₃ C₃₁

Properties of determinants state that if A is a square matrix of the order n, then Det (A) is the sum of products of elements of a row (or a column) with the corresponding cofactor of that element.

$\left|A\right|=a_{11}{C}_{11}+a_{21}{C}_{21}+a_{31\text{'}}{C}_{31}\left[\mathrm{Calculating}\mathrm{along}{C}_1\right]$