CBSE Sample Paper 2020-21 · Part A: Section I · 1 mark

Q6: Value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$

Sample 2020-21Ch 4 · Determinants

The question

Let $A = [a_{ij}]$ be a square matrix of order 3×3 and $|A| = -7$ ⁠. Find the value of

a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}

where $A_{ij}$ is the cofactor of element $a_{ij}$

Exam answer

The sum of the products of the elements of a row with the cofactors of the corresponding elements of another row is zero.

Why?

Here the elements come from row 1, but the cofactors $A_{21}, A_{22}, A_{23}$ belong to row 2.

Expanding along row 2 gives $a_{21}A_{21} + a_{22}A_{22} + a_{23}A_{23}$ ⁠. The cofactors of row 2 do not use row 2's own entries. So putting row 1's entries in their place gives the expansion of

\begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{11} & a_{12} & a_{13} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} = 0

This determinant has two identical rows, so it is 0.

a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0

Why?

Watch out: $|A| = -7$ is not needed. It is the answer only when a row is multiplied by its own cofactors:

\begin{aligned} &a_{11}A_{11} + a_{12}A_{12} \\ &\quad + a_{13}A_{13} = |A| \end{aligned}

$\therefore$ the required value is $0$ ⁠.

Where this comes from

Chapter 4, Determinants: Cofactors: one row's elements, another's cofactors.

Quick lesson · Worked NCERT solutions · Interactive lesson

Found an error?

Tell us if something on this page is wrong or unclear. Send feedback on this question →

Determinants in past papers: question 32 of 34 · see them all

Q6: Value of a11A21+a12A22+a13A23a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}a11​A21​+a12​A22​+a13​A23​

Q6: Value of $a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23}$