Matrices — JEE Advanced PYQs

Which one of the following matrices can be obtained by performing elementary row transformations on the $3\times 3$ identity matrix ?

Consider the matrix

Let $\mathrm{p},\mathrm{q},\mathrm{r},\mathrm{s},\mathrm{a},\mathrm{b},\mathrm{c}$ and d be integers such that …

Let $\mathbb{R}$ denote the set of all real numbers and let $\mathrm{i}=\sqrt{-1}$. Consider the matrices
$\mathrm{S}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ and $\mathrm{T}=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$.
Let …

Consider the matrix

Let the transpose of a matrix $X$ be denoted by ${\mathrm{X}}^T$. Then the number of 3 × 3 invertible matrices Q with integer entries, such that …

Let $I=\left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right)$ and $P=\left(\begin{array}{ll}2& 0\\ 0& 3\end{array}\right)$. Let $Q=\left(\begin{array}{ll}\mathrm{x}& \mathrm{y}\\ \mathrm{z}& 4\end{array}\right)$ for some non-zero real numbers $x,y$, and $z$, fo …

Let $\alpha$ and $\beta$ be the distinct roots of the equation ${\mathrm{x}}^2+\mathrm{x}-1=0$. Consider the set $\mathrm{T}=\{1,\alpha ,\beta \}$. For a $3\times 3$ matrix $\mathrm{M}={\left({\mathrm{a}}_{\mathrm{ij}}\right)}_{3\times 3}$, define …

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

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

Let $\alpha ,\beta$ and $\gamma$ be real numbers. consider the following system of linear equations

Match each entry in List - I to the correct entries in List-II

List-I
(P) If …

Let $p,q,r$ be nonzero real numbers that are, respectively, the $10^{\text{th }},100^{\text{th }}$ and $1000^{\text{th }}$ terms of a harmonic progression. Consider the system of linear equations




| List-I | | Lis …

Let $\beta$ be a real number. Consider the matrix

If $A^7-(\beta -1)A^6-\beta A^5$ is a singular matrix, then the value of $9\beta$ is ____.

If $M=\left(\begin{array}{cc}\frac{5}{2}& \frac{3}{2}\\ -\frac{3}{2}& -\frac{1}{2}\end{array}\right)$, then which of the following matrices is equal to $M^{2022}$?

For any 3 × 3 matrix M, let $|M|$ denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If $G=(I-EF{)}^{-1}$, then which of the following statements is (are) TRUE?

Let M be a $3\times 3$ invertible matrix with real entries and let I denote the $3\times 3$ identity matrix. If $M^{-1}=\text{adj}(\text{adj }M)$, then which of the following statement is/are ALWAYS TRUE?

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a 2 × 2 matrix such that the trace of $A$ is 3 and the trace of $A^3$ is -18, then the value of the determinant of $A$ is ____

Let $\mathrm{M}=\left[\begin{array}{cc}{\sin }^4\theta & -1-{\sin }^2\theta \\ 1+{\cos }^2\theta & {\cos }^4\theta \end{array}\right]=\alpha \mathrm{I}+\beta {\mathrm{M}}^{-1}$,
where $\alpha =\alpha (\theta )$ and $\beta =\beta (\theta )$ are real number, and I is the $2\times 2$ id …

Let $\mathrm{M}=\left[\begin{array}{lll}0& 1& \mathrm{a}\\ 1& 2& 3\\ 3& \mathrm{b}& 1\end{array}\right]$ and $\mathrm{adj}\mathrm{M}=\left[\begin{array}{ccc}-1& 1& -1\\ 8& -6& 2\\ -5& 3& -1\end{array}\right]$ where a and b are real numbers. Which of the fo …

Let $P_1=I=\left[\begin{array}{lll}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$, $P_2=\left[\begin{array}{lll}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right]$, $P_3=\left[\begin{array}{lll}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$, …

Let $\mathrm{x}\in \mathbb{R}$ and let $\mathrm{P}=\left[\begin{array}{lll}1& 1& 1\\ 0& 2& 2\\ 0& 0& 3\end{array}\right]$, $\mathrm{Q}=\left[\begin{array}{ccc}2& \mathrm{x}& \mathrm{x}\\ 0& 4& 0\\ \mathrm{x}& \mathrm{x}& 6\end{array}\right]$ and …

Let $S$ be the set of all column matrices $\left[\begin{array}{l}{b}_1\\ b_2\\ b_3\end{array}\right]$ such that $b_1,b_2,b_3\in \mathbb{R}$ and the system of equations (in real variables) …

Which of the following is(are) NOT the square of a $3\times 3$ matrix with real entries?

For a real number $\alpha$, if the system

of lin …

How many $3\times 3$ matrices $M$ with entries from $\{0,1,2\}$ are there, for which the sum of the diagonal entries of $M^{T}M$ is 5?

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha \in \mathbb{R}$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that $PQ=kI$, where $k\in \mathbb{R},k\ne 0$ and $I$ is the identity matrix of order 3. If …

Let $P=\left[\begin{array}{ccc}1& 0& 0\\ 4& 1& 0\\ 16& 4& 1\end{array}\right]$ and $I$ be the identity matrix of order 3. If $Q=\left[q_{ij}\right]$ is a matrix such that $P^{50}-Q=I$, then $\frac{q_{31}+q_{32}}{q_{21}}$ equals

Let $a,\lambda ,\mu \in \mathbb{R}$. Consider the system of linear equations

Which of the following statement(s) is(are) correct?

Let $X$ and $Y$ be two arbitrary, $3\times 3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times 3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Let $M$ and $N$ be two $3\times 3$ matrices such that $MN=NM$. Further, if $M\ne N^2$ and $M^2=N^4$, then

Let $M$ be a 2 × 2 symmetric matrix with integer entries. Then $M$ is invertible if

For $3\times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct ?

If $P$ is a $3\times 3$ matrix such that $P^T=2P+I$, where $P^T$ is the transpose of $P$ and $I$ is the $3\times 3$ identity matrix, then there exists a column matrix …

If the adjoint of a $3\times 3$ matrix $P$ is $\left[\begin{array}{lll}1& 4& 4\\ 2& 1& 7\\ 1& 1& 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are)

Let $a,b$ and $c$ be three real numbers satisfying

If the point $P(a,b,c)$, wi …

Let $M$ be a $3\times 3$ matrix satisfying …

Let k be a positive real number and let …

The number of 3×3 matrices A whose entries are either 0 or 1 and for which the system $A\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}1\\ 0\\ 0\end{array}\right]$ has exactly two distinct solutions, is

Let p be an odd prime number and $T_p$ be the following set of $2\times 2$ matrices:


The number of A in $T_p$ such that A is either symmetric or skew-symm …

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:

Then the number of such points for which $x^2+y^2+z^2\le 100$ is

Let $\mathcal{A}$ be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

The number of matrices in $\mathcal{A}$ is

Let $\mathcal{A}$ be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

The number of matrices $A$ in $\mathcal{A}$ for which the system of linear equations …

Consider the system of equations


STATEMENT-1 : The system of equations has no solution for $k\ne 3$.

and

STATEMENT-2 : The determinant …

Consider the system of equations
$ax+by=0,cx+dy=0$, where $a,b,c,d\in \{0,1\}$.

STATEMENT-1 : The probability that the system of equations has a unique solution is $\frac{3}{8}$.

and

STATEMENT-2 : The probability that the system of equations has a solution is 1 .