CBSE Board Examination 2023 · Section A · 1 mark

Q2: $|4A^{-1}|$ for a $2 \times 2$ matrix with $|A| = 2$

The question

If $|A| = 2$ ⁠, where A is a $2 \times 2$ matrix, then $|4A^{-1}|$ equals :

(a) $4$

(b) $2$

(d) $\frac{1}{32}$

Exam answer

For a matrix of order 2, $|kB| = k^2|B|$ ⁠, and $|A^{-1}| = \frac{1}{|A|}$ ⁠.

Why?

Multiplying a matrix by $k$ multiplies every row by $k$ ⁠. Each row gives one factor $k$ outside the determinant, so for order $n$ ⁠:

|kB| = k^n|B|

For the inverse, take determinants of $AA^{-1} = I$ ⁠:

\begin{aligned} |A|\,|A^{-1}| &= |I| = 1 \\ |A^{-1}| &= \frac{1}{|A|} \end{aligned}

(Divide both sides by $|A|$ ⁠.) With $|A| = 2$ this is $\frac12$ ⁠.

|A^{-1}| = \frac{1}{|A|} = \frac{1}{2}

\begin{aligned} |4A^{-1}| &= 4^2\,|A^{-1}| \\ &= 16 \times \frac{1}{2} = 8 \end{aligned}

Why?

Here $B = A^{-1}$ ⁠, $k = 4$ and the order is $n = 2$ ⁠, so the factor is $4^2 = 16$ ⁠.

\begin{aligned} |4A^{-1}| &= 4^2 \times \frac12 \\ &= 16 \times \frac12 \\ &= \frac{16}{2} = 8 \end{aligned}

Watch out: $|4A^{-1}|$ is not $4|A^{-1}| = 2$ ⁠, which is option (b). The power of $k$ is the order of the matrix.

Answer: (c) $8$

Where this comes from

Chapter 4, Determinants: Determinant of kA and of the inverse.

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 $|A| = 2$ and the order 2 are given: $|A^{-1}| = \frac12$ ⁠, and $|4A^{-1}| = 4^2 \cdot \frac12 = 8$ ⁠.

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Determinants in past papers: question 11 of 34 · see them all

Q2: ∣4A−1∣|4A^{-1}|∣4A−1∣ for a 2×22 \times 22×2 matrix with ∣A∣=2|A| = 2∣A∣=2