Mathematics · Matrices

JEE Advanced 2023 — Paper 2 — Question 5

Let $\mathrm{M}=\left(a_{i j}\right), i, j \in\{1,2,3\}$ , be the $3 \times 3$ matrix such that $a_{i j}=1$ if $j+1$ is divisible by $i$ , otherwise

aij $=0$ . Then which of the following statements is(are) true?

Option A:
$M$ is invertible
Option B:
There exists a nonzero column matrix $\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)$ and such that $M\left(\begin{array}{l}a_{1} \\ a_{2} \\ a_{3}\end{array}\right)=\left(\begin{array}{l}-a_{1} \\ -a_{2} \\ -a_{3}\end{array}\right)$
Correct
Option C:
The set $\left\{X \in \mathbb{R}^{3}: M X=0\right\} \neq\{\mathbf{0}\}$ , where $\mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$
Correct
Option D:
The matrix $(M-2 I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix

Answer: B, C

Step-by-step solution

$M=\left[\begin{array}{lll}1 & 1 & 1\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$ $\because|\mathrm{M}|=0$ so matrix is non-invertible so option A is incorrect $\left[\begin{array}{lll}1 & 1 & 1\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]\left[\begin{array}{l}a_{1}\\ a_{2}\\ a_{3}\end{array}\right]=\left[\begin{array}{c}a_{1}+a_{2}+a_{3}\\ a_{1}+a_{3}\\ a_{2}\end{array}\right]=\left[\begin{array}{l}-a_{1}\\ -a_{2}\\ -a_{3}\end{array}\right]$ $2 a_{1}+a_{2}+a_{3}=0, a_{2}+a_{1}+a_{3}=0, a_{2}+a_{3}=0, a_{1}=0 \Rightarrow a_{2}=-a_{3}$ So $B$ is correct $a_{1}+a_{2}+a_{3}=0, a_{1}+a_{3}=0, a_{2}=0 \Rightarrow a_{1}=-a_{3}$ So C is correct

\begin{aligned} |M-2 I| & =\left|\begin{array}{ccc} -1 & 1 & 1 \\ 1 & -2 & 1 \\ 0 & 1 & -2 \end{array}\right| \\& =(-1) 3-(-2)+1 \times 1 \\& =-3+2+1=0 \end{aligned}

So $\mathrm{M}-2 \mathrm{I}$ is non-invertible.

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2023
Paper: Paper 2
Subject: Mathematics
Chapter: Matrices
Topic: Inverse of a Matrix