If A = |(a11,a12,a13)(a21,a22,a23)(a31,a32,a33)| and Cij is cofactor of aij in A,

Question

If A = \(\begin{vmatrix} a_{11} & a_{12} & a_{13} \\[0.3em] a_{21} & a_{22} & a_{23} \\[0.3em] a_{31} &a_{32} & a_{33} \end{vmatrix} \) and C_ij is cofactor of a_ij in A, then value of |A| is given by

A. a₁₁C₃₁ + a₁₂C₃₂ + a₁₃C₃₃ 

B. a₁₁C₁₁ + a₁₂C₂₁ + a₁₃C₃₁ 

C. a₂₁C₁₁ + a₂₂C₁₂ + a₂₃C₁₃ 

D. a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁

Answer 1

Let us understand what cofactor of an element is. 

A cofactor is the number you get when you remove the column and row of a designated element in a matrix, which is just a numerical grid in the form of a rectangle or a square. 

The cofactor is always preceded by a positive (+) or negative (-) sign, depending whether the element is in a + or - position. 

It is \(\begin{bmatrix} + &-& + \\[0.3em] - & + &- \\[0.3em] + & - & + \end{bmatrix}\)

Let us recall how to find the cofactor of any element : 

If we are given with,

\(\begin{bmatrix} a_{11} &a_{12}& a_{13} \\[0.3em] a_{21} & a_{22} &a_{23} \\[0.3em] a_{31} & a_{32} & a_{33} \end{bmatrix}\)

Cofactor of any element say a₁₁ is found by eliminating first row and first column.

Cofactor of a_{11 =} \(\begin{bmatrix} a_{22} &a_{23} \\[0.3em] a_{32} & a_{33} \end{bmatrix}\)

⇒ Cofactor of a₁₂ = a₂₁ × a₃₃ – a₂₃ × a₃₁ 

The sign of cofactor of a₁₂ is (-). 

We are given that,

Thus, proved.