Mark the correct alternative in the following:

If and C_ij is cofactor of a_ij in A, then value of |A| is given by

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

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

If we are given with,

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

⇒ Cofactor of a₁₁ = a₂₂ × a₃₃ – a₂₃ × a₃₂

The sign of cofactor of a₁₁ is (+).

And, cofactor of any element, say a₁₂ is found by eliminating first row and second column.

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

The sign of cofactor of a₁₂ is (-).

We are given that,

And C_ij is the cofactor of a_ij in A.

Determinant of 3 × 3 matrix is given as,

Or,

Or using the definition of cofactors,

Thus, proved.