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Matrices and Determinants question

2023 · Shift 2 · Q22

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JEE Advanced
Mathematics
Matrices and Determinants
2023 · Shift 2 · Q22

Matrices and Determinants question

2023 · Shift 2 · Q22

JEE AdvancedMathematicsMatrices and DeterminantsMultiple correct+4 / −2

Let

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

a_{i j}=0

. Then which of the following statements is(are) true?

A
$M$ is invertible
B
There exists a nonzero column matrix $\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)$ such that $M\left(\begin{array}{l}a_1 \\ a_2 \\ a_3\end{array}\right)=\left(\begin{array}{l}-a_1 \\ -a_2 \\ -a_3\end{array}\right)$
C
The set $\left\{X \in \mathbb{R}^3: M X=\mathbf{0}\right\} eq\{\mathbf{0}\}$ , where $\mathbf{0}=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$
D
The matrix $(M-2 I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix

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