Q.If $\Delta = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$ and $A_{ij}$ is Cofactors of $a_{ij}$, then value of $\Delta$ is given by (A) $a_{11} A_{31} + a_{12} A_{32} + a_{13} A_{33}$ (B) $a_{11} A_{11} + a_{12} A_{21} + a_{13} A_{31}$ (C) $a_{21} A_{11} + a_{22} A_{12} + a_{23} A_{13}$ (D) $a_{11} A_{11} + a_{21} A_{21} + a_{31} A_{31}$

Concept: Determinant Evaluation Using Cofactor Expansion — the determinant equals the sum of the products of the elements of any row (or column) with their corresponding cofactors.

Step 1: The standard expansion along the first row is

$\Delta = a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$.

Step 2: Option (D) gives $a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$ — this is expansion along the first column, which is equally valid. …

The question tests a core property of determinants: the cofactor expansion (also called Laplace expansion). A determinant is not just a number — it is built from a structured sum that respects rows and columns. The cofactor $A_{ij}$ carries a sign $(-1)^{i+j}$ times the minor (the determinant left after deleting row $i$ and column $j$). The key identity is:

This works because each term pairs an element with its cofactor, and the signs and minors reconstruct the full determinant. If you mix a row’s elements with cofactors from a different row, you get zero — that’s the “wrong” expansion.

Let’s check each option.

1. Option (A): $a_{11}A_{31} + a_{12}A_{32} + a_{13}A_{33}$

   Here the elements are from row 1, but the cofactors are from row 3. This is an expansion using a different row’s cofactors — the result is always zero, not $\Delta$. So (A) is wrong.

2. Option (B): $a_{11}A_{11} + a_{12}A_{21} + a_{13}A_{31}$

   The elements are from row 1, but the cofactors are from column 1 (first subscript varies: $A_{11}, A_{21}, A_{31}$). This is a mismatched mix — not a valid row or column expansion. It does not equal $\Delta$ in general. So (B) is wrong.

3. Option (C): $a_{21}A_{11} + a_{22}A_{12} + a_{23}A_{13}$

   Elements from row 2, cofactors from row 1. Again, a “wrong row” expansion — gives zero. So (C) is wrong.

4. Option (D): $a_{11}A_{11} + a_{21}A_{21} + a_{31}A_{31}$ …

Method: Recognising a Valid Row/Column Cofactor Expansion

Use this to check (or answer an MCQ about) whether a given "element times cofactor" sum actually equals the determinant.

Steps

Step 1: Recall the only two valid templates

• Row expansion (fixed row $i$): $\Delta = a_{i1}A_{i1}+a_{i2}A_{i2}+a_{i3}A_{i3}$ — elements and cofactors both come from row $i$.
• Column expansion (fixed column $j$): $\Delta = a_{1j}A_{1j}+a_{2j}A_{2j}+a_{3j}A_{3j}$ — elements and cofactors both come from column $j$.

Step 2: Check whether the given expression matches one of these templates exactly

Verify that the row (or column) index of every element matches the row (or column) index of its paired cofactor consistently across all three terms. …

Common Mistakes

Mistake 1: Assuming any "element times cofactor" sum automatically equals the determinant

Why it's wrong: only sums where the row/column index of every element matches the row/column index of its paired cofactor, consistently across all three terms, actually equal $\Delta$ — an arbitrary mix of indices generally does not. Correct approach: explicitly check each option against the two valid templates (fixed-row or fixed-column expansion) before choosing.

Mistake 2: Confusing a genuine column expansion with a mismatched index pattern …