CBSE Sample Paper 2022-23 · Section A · 1 mark

Q2: $|AA'|$ when $|A'| = -3$

Sample 2022-23Ch 4 · Determinants Ch 3 · Matrices

The question

If A is a square matrix of order 3, $|A'| = -3$ ⁠, then $|AA'| =$

(a) $9$

(b) $-9$

(d) $-3$

Exam answer

$|A| = |A'| = -3$

Why?

A square matrix and its transpose have the same determinant:

|A'| = |A|

So $|A| = -3$ as well.

\begin{aligned} |AA'| &= |A|\,|A'| \\ &= (-3)(-3) = 9 \end{aligned}

Why?

The determinant of a product of two square matrices of the same order is the product of their determinants:

|PQ| = |P|\,|Q|

Watch out: two negatives multiply to a positive, so the value is $9$ ⁠, not $-9$ (option (b)). The order 3 is not needed here; it would matter only for a multiple such as $|kA| = k^3|A|$ ⁠.

Answer: (a) $9$

Where this comes from

Chapter 4, Determinants: Determinant of a product and of a transpose.

Question type: One-mark questions on $|kA|$ ⁠, $|\operatorname{adj}A|$ and $|A^{-1}|$

How to spot it: click to see

Type 10. One-mark questions on $|kA|$ ⁠, $|\operatorname{adj}A|$ and $|A^{-1}|$

Spot it: an MCQ about a matrix you are not given, only its order.

Here: Only $|A'| = -3$ and the order are given: $|A| = |A'| = -3$ ⁠, and $|AA'| = |A||A'| = 9$ ⁠.

Quick lesson · Worked NCERT solutions · Interactive lesson

Also uses Chapter 3, Matrices.

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Determinants in past papers: question 28 of 34 · see them all

Q2: ∣AA′∣|AA'|∣AA′∣ when ∣A′∣=−3|A'| = -3∣A′∣=−3

Q2: $|AA'|$ when $|A'| = -3$