Matrices & Determinants · JEE Advanced previous year questions

2023 2 questions

1. Q52023 · 4 Jun 2023 · Paper 2· One or more correct

   Let $M=(a_{ij})$, $i,j\in \{1,2,3\}$, be the $3\times 3$ matrix such that $a_{ij}=1$ if $j+1$ is divisible by $i$, otherwise $a_{ij}=0$. Then which of the following statements is(are) true?

   1. A.$M$ is invertible
   2. B.There exists a nonzero column matrix $\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right)$ such that $M\left(\begin{array}{c}{a}_1\\ a_2\\ a_3\end{array}\right)=\left(\begin{array}{c}-a_1\\ -a_2\\ -a_3\end{array}\right)$
   3. C.The set $\{X\in {\mathbb{R}}^3:MX=0\}\ne \{0\}$, where $0=\left(\begin{array}{c}0\\ 0\\ 0\end{array}\right)$
   4. D.The matrix $(M-2I)$ is invertible, where $I$ is the $3\times 3$ identity matrix
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   Answer: B, C

2. Q122023 · 4 Jun 2023 · Paper 2· Numerical answer

   Let $R=\left\{\left(\begin{array}{ccc}a& 3& b\\ c& 2& d\\ 0& 5& 0\end{array}\right):a,b,c,d\in \{0,3,5,7,11,13,17,19\}\right\}$. Then the number of invertible matrices in $R$ is ____.

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   Answer: 3780