Matrices & Determinants · JEE Advanced previous year questions

Consider the matrix $$P=\left(\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array}\right).$$ Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3\times 3$ invertible matrices $Q$ with integer entries, such that $$Q^{-1}=Q^T\text{and}PQ=QP,$$ is

1. A.32
2. B.8
3. C.16
4. D.24

Let $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ and $P=\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)$. Let $Q=\left(\begin{array}{cc}x& y\\ z& 4\end{array}\right)$ for some non-zero real numbers $x$, $y$, and $z$, for which there is a $2\times 2$ matrix $R$ with all entries being non-zero real numbers, such that $QR=RP$. Then which of the following statements is (are) TRUE?

1. A.The determinant of $Q-2I$ is zero
2. B.The determinant of $Q-6I$ is 12
3. C.The determinant of $Q-3I$ is 15
4. D.$yz=2$

Let $S=\left\{A=\left(\begin{array}{ccc}0& 1& c\\ 1& a& d\\ 1& b& e\end{array}\right):a,b,c,d,e\in \{0,1\}\text{ and }|A|\in \{-1,1\}\right\}$, where $|A|$ denotes the determinant of $A$. Then the number of elements in $S$ is _____.

Let $\alpha$ and $\beta$ be the distinct roots of the equation $x^2+x-1=0$. Consider the set $T=\{1,\alpha ,\beta \}$. For a $3\times 3$ matrix $M=(a_{ij}{)}_{3\times 3}$, define $R_i=a_{i1}+a_{i2}+a_{i3}$ and $C_j=a_{1j}+a_{2j}+a_{3j}$ for $i=1,2,3$ and $j=1,2,3$. Match each entry in List-I to the correct entry in List-II. List-I: (P) The number of matrices $M=(a_{ij}{)}_{3\times 3}$ with all entries in $T$ such that $R_i=C_j=0$ for all $i,j$, is (Q) The number of symmetric matrices $M=(a_{ij}{)}_{3\times 3}$ with all entries in $T$ such that $C_j=0$ for all $j$, is (R) Let $M=(a_{ij}{)}_{3\times 3}$ be a skew symmetric matrix such that $a_{ij}\in T$ for $i>j$. Then the number of elements in the set $\left\{\left(\begin{array}{c}x\\ y\\ z\end{array}\right):x,y,z\in \mathbb{R},M\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}{a}_{12}\\ 0\\ -a_{23}\end{array}\right)\right\}$ is (S) Let $M=(a_{ij}{)}_{3\times 3}$ be a matrix with all entries in $T$ such that $R_i=0$ for all $i$. Then the absolute value of the determinant of $M$ is List-II: (1) 1 (2) 12 (3) infinite (4) 6 (5) 0 The correct option is

1. A.(P) $\to$ (4) \quad (Q) $\to$ (2) \quad (R) $\to$ (5) \quad (S) $\to$ (1)
2. B.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (1) \quad (S) $\to$ (5)
3. C.(P) $\to$ (2) \quad (Q) $\to$ (4) \quad (R) $\to$ (3) \quad (S) $\to$ (5)
4. D.(P) $\to$ (1) \quad (Q) $\to$ (5) \quad (R) $\to$ (3) \quad (S) $\to$ (4)

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

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 ____.