If \(A=\begin{vmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{vmatrix}\) and \(A_{ij}\) is cofactors of \(a_{ij}\), then value of \(|A|\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}\)
A determinant equals the sum of the products of the elements of any one row or column with their own cofactors. This is the expansion of a determinant along that row or column.
Expanding along the first column, whose elements are \(a_{11}, a_{21}, a_{31}\):
\(|A| = a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}\)
Why the other options are incorrect
Option (1) \(a_{11}A_{31}+a_{12}A_{32}+a_{13}A_{33}\) — Incorrect. The elements of row 1 are multiplied by the cofactors of row 3. The sum of the products of the elements of one row with the cofactors of another row is always 0.
Option (2) \(a_{11}A_{11}+a_{12}A_{21}+a_{13}A_{31}\) — Incorrect. Only the first term pairs an element with its own cofactor; \(a_{12}\) is paired with \(A_{21}\) instead of \(A_{12}\), so this is not an expansion of |A|.
Option (3) \(a_{21}A_{11}+a_{22}A_{12}+a_{23}A_{13}\) — Incorrect. The elements of row 2 are multiplied by the cofactors of row 1, so the sum is 0.
NCERT (2026-27 edition):
Minors and Cofactors
