Matrices — JEE Main PYQs

Let $A$ be a $3\times 3$ matrix such that $|\text{adj}(\text{adj}(\text{adj }A))|=81$. If
$S=\{n\in \mathbb{Z}:(|\text{adj}(\text{adj }A)|{)}^{\frac{(n-1{)}^2}{2}}=|A{|}^{(3n^2-5n-4)}\}$, then ${\sum }_{n\in S}|A^{(n^2+n)}|$ is equal to

The number of singular matrices of order 2, whose elements are from the set $\{2,3,6,9\}$ is

Let A be a $3\times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=3\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$. Then the maximum value of $\det (A)$ is ____.

Let A be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying $A^2+xA+yI=O$ lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is $e$ and the length of the latus rec …

Let $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ and $B=I+\mathrm{adj}(A)+(\mathrm{adj}A{)}^2+\dots +(\mathrm{adj}A{)}^{10}$. Then, the sum of all the elements of the matrix $B$ is

Let $A$ be a $2\times 2$ symmetric matrix such that $A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 7\end{array}\right]$ and the determinant of $A$ be 1. If $A^{-1}=\alpha A+\beta I$, where $I$ is the identity matrix of order $2\times 2$, then $\alpha +\beta$ equal …

The values of $m,n$, for which the system of equations $x+y+z=4$, $2x+5y+5z=17$, $x+2y+mz=n$ has infinitely many solutions, satisfy the equation :

If $A$ is a square matrix of order 3 such that $\det (A)=3$ and $\det \left(\mathrm{adj}\left(-4\mathrm{adj}.\left(-3\mathrm{adj}\left(3\mathrm{adj}\left((2A{)}^{-1}\right)\right)\right)\right)\right)=2^m{3}^n$. Then $m+2n$ is equ …

Let $\mathrm{A}=\left[\begin{array}{lll}2& \mathrm{a}& 0\\ 1& 3& 1\\ 0& 5& \mathrm{b}\end{array}\right]$. If ${\mathrm{A}}^3=4{\mathrm{A}}^2-\mathrm{A}-21\mathrm{I}$, where I is the identity matrix of order $3\times 3$, then $2\mathrm{a}+3\mathrm{b}$ is eq …

Let $A=\left[\begin{array}{cc}2& -1\\ 1& 1\end{array}\right]$. If the sum of the diagonal elements of $A^{13}$ is $3^n$, then n is equal to ______.

If the system of equations $\mathrm{x}+4\mathrm{y}-\mathrm{z}=\lambda$, $7\mathrm{x}+9\mathrm{y}+\mu \mathrm{z}=-3$, $5\mathrm{x}+\mathrm{y}+2\mathrm{z}=-1$ has infinitely many solutions, then $(2\mu +3\lambda )$ is equal to :

Let $\lambda ,\mu \in \mathrm{R}$. If the system of equations
$3\mathrm{x}+5\mathrm{y}+\lambda \mathrm{z}=3$
$7\mathrm{x}+11\mathrm{y}-9\mathrm{z}=2$
$97\mathrm{x}+155\mathrm{y}-189\mathrm{z}=\mu$
has infinitely many solutions, then $\mu +2\lambda$ is equal to :

Let A be a non-singular matrix of order 3. If $\det (3\mathrm{adj}(2\mathrm{adj}((\det A)A)))=3^{-13}\cdot 2^{-10}$ and $\det (3\mathrm{adj}(2A))=2^m\cdot 3^n$, then $|3m+2n|$ is equal to

Let $\mathrm{B}=\left[\begin{array}{ll}1& 3\\ 1& 5\end{array}\right]$ and A be a $2\times 2$ matrix such that ${\mathrm{AB}}^{-1}={\mathrm{A}}^{-1}$. If ${\mathrm{BCB}}^{-1}=\mathrm{A}$ and ${\mathrm{C}}^4+\alpha {\mathrm{C}}^2+\beta \mathrm{I}=\mathrm{O}$, then …

Consider the matrices: $\mathrm{A}=\left[\begin{array}{cc}2& -5\\ 3& \mathrm{m}\end{array}\right],\mathrm{B}=\left[\begin{array}{c}20\\ \mathrm{m}\end{array}\right]$ and $\mathrm{X}=\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\end{array}\right]$. Let the set of all m …

Let $\mathrm{A}=\left[\begin{array}{ccc}2& 1& 2\\ 6& 2& 11\\ 3& 3& 2\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{ccc}1& 2& 0\\ 5& 0& 2\\ 7& 1& 5\end{array}\right]$. The sum of the prime factors of …

Let a, b, c be in A.P. and the line $ax+by+c=0$ passes through a fixed point P. For the system of equations

to have infinite solutions, let the point be Q$(\alpha ,\beta )$. T …

Let A be a $3\times 3$ real matrix such that …

Consider the system of linear equation $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4{\lambda }^{2}z={\mu }^2+15$, where $\lambda ,\mu \in \mathrm{R}$. Which one of the following statements is NOT correct?

Let A be a $2\times 2$ real matrix and I be the identity matrix of order 2 . If the roots of the equation $|A-xI|=0$ be -1 and 3 , then the sum of the diagonal elements of the matrix ${\mathrm{A}}^2$ is ______

Consider the matrix $\mathrm{f}(\mathrm{x})=\left[\begin{array}{ccc}\cos \mathrm{x}& -\sin \mathrm{x}& 0\\ \sin \mathrm{x}& \cos \mathrm{x}& 0\\ 0& 0& 1\end{array}\right]$.

Given below are two statements :
Statement I: $\mathrm{f}(-\mathrm{x})$ is the inverse of the …

Let $A=\left[\begin{array}{ccc}2& 0& 1\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]$, $B=[B_1,B_2,B_3]$, where $B_1$, $B_2$, $B_3$ are column matrices, and $A{B}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, $A{B}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right]$, …

Let the system of equations $x+2y+3z=5$, $2x+3y+z=9,4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda +2\mu$ is equal to :

Let $\mathrm{A}={\mathrm{I}}_2-2{\mathrm{MM}}^{\mathrm{T}}$, where M is real matrix of order $2\times 1$ such that the relation ${\mathrm{M}}^{\mathrm{T}}\mathrm{M}={\mathrm{I}}_1$ holds. If $\lambda$ is a real number such that the relation …

If the system of linear equations
$x-2y+z=-4$
$2\mathrm{x}+\alpha \mathrm{y}+3\mathrm{z}=5$
$3x-y+\beta z=3$
has infinitely many solutions, then $12\alpha +13\beta$ is equal to

If $A=\left[\begin{array}{cc}\sqrt{2}& 1\\ -1& \sqrt{2}\end{array}\right],B=\left[\begin{array}{ll}1& 0\\ 1& 1\end{array}\right],C=AB{A}^T$ and $X=A^T{C}^{2}A$, then $\det X$ is equal to :

If the system of equations
$2x+3y-z=5$
$x+\alpha y+3z=-4$
$3x-y+\beta z=7$
has infinitely many solutions, then $13\alpha \beta$ is equal to

Let A be a square matrix such that ${\mathrm{AA}}^{\mathrm{T}}=\mathrm{I}$. Then $\frac{1}{2}\mathrm{A}\left[{\left(\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}\right)}^2+{\left(\mathrm{A}-{\mathrm{A}}^{\mathrm{T}}\right)}^2\right]$ is equal to

Consider the system of linear equations $x+y+z=5,x+2y+{\lambda }^{2}z=9$, $\mathrm{x}+3\mathrm{y}+\lambda \mathrm{z}=\mu$, where $\lambda ,\mu \in \mathrm{R}$. Then, which of the following statement is NOT correct?

Let $\mathrm{R}=\left(\begin{array}{ccc}\mathrm{x}& 0& 0\\ 0& \mathrm{y}& 0\\ 0& 0& \mathrm{z}\end{array}\right)$ be a non-zero $3\times 3$ matrix, where …

For the system of equations
$x+y+z=6$
$x+2y+\alpha z=10$
$x+3y+5z=\beta$, which one of the following is NOT true?

Let P be a square matrix such that $P^2=I-P$. For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $P^{\alpha }+P^{\beta }=\gamma I-29P$ and $P^{\alpha }-P^{\beta }=\delta I-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to

Let $\mathrm{P}=\left[\begin{array}{cc}\frac{\sqrt{3}}{2}& \frac{1}{2}\\ -\frac{1}{2}& \frac{\sqrt{3}}{2}\end{array}\right],\mathrm{A}=\left[\begin{array}{ll}1& 1\\ 0& 1\end{array}\right]$ and $\mathrm{Q}={\mathrm{PAP}}^{\mathrm{T}}$. If …

Let $\mathrm{A}=\left[\begin{array}{ccc}2& 1& 0\\ 1& 2& -1\\ 0& -1& 2\end{array}\right]$. If $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}2\mathrm{A}))|=(16{)}^{\mathrm{n}}$, then n is equal to

Let $A={\left[a_{ij}\right]}_{2\times 2}$ where $a_{ij}\ne 0$ for all $i,j$ and $A^2=I$. Let a be the sum of all diagonal elements of A and $\mathrm{b}=|\mathrm{A}|$, then $3{\mathrm{a}}^2+4{\mathrm{b}}^2$ is equal to

If the system of equations



has infinitely many solutions, then $2\mathrm{a}+3\mathrm{b}$ is equal to

If $\mathrm{A}=\left[\begin{array}{cc}1& 5\\ \lambda & 10\end{array}\right]$, ${\mathrm{A}}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$ and $\alpha +\beta =-2$, then $4{\alpha }^2+{\beta }^2+{\lambda }^2$ is equal to:

Let S be the set of values of $\lambda$, for which the system of equations
$6\lambda \mathrm{x}-3\mathrm{y}+3\mathrm{z}=4{\lambda }^2$,
$2\mathrm{x}+6\lambda \mathrm{y}+4\mathrm{z}=1$,
$3x+2y+3\lambda z=\lambda$ has no solution. Then …

If A is a $3\times 3$ matrix and $|\mathrm{A}|=2$, then $\left|3adj\left(|3A|A^2\right)\right|$ is equal to

For the system of linear equations


Which of the following is NOT correct?

Let $\mathrm{A}=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$. If $\mathrm{B}=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]\mathrm{A}\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix …

If the system of equations
$2x+y-z=5$
$2x-5y+\lambda z=\mu$
$x+2y-5z=7$
has infinitely many solutions, then $(\lambda +\mu {)}^2+(\lambda -\mu {)}^2$ is equal to

For the system of linear equations
$2\mathrm{x}+4\mathrm{y}+2\mathrm{az}=\mathrm{b}$
$x+2y+3z=4$
$2x-5y+2z=8$
which of the following is NOT correct?

The number of symmetric matrices of order 3 , with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$, is :

Let $B=\left[\begin{array}{ccc}1& 3& \alpha \\ 1& 2& 3\\ \alpha & \alpha & 4\end{array}\right]$, $\alpha >2$ be the adjoint of a matrix A and $|A|=2$, then $\left[\begin{array}{ccc}\alpha & -2\alpha & \alpha \end{array}\right]B\left[\begin{array}{c}\alpha \\ -2\alpha \\ \alpha \end{array}\right]$ is equal to

Let the determinant of a square matrix A of order m be $\mathrm{m}-\mathrm{n}$, where m and n satisfy $4\mathrm{m}+\mathrm{n}=22$ and $17\mathrm{m}+4\mathrm{n}=93$. If det $(\mathrm{n}\mathrm{adj}(\mathrm{adj}(\mathrm{mA})))=3^a{5}^b{6}^c$, then …

Let the system of linear equations
$-x+2y-9z=7$
$-x+3y+7z=9$
$-2x+y+5z=8$
$-3x+y+13z=\lambda$
has a unique solution $\mathrm{x}=\alpha ,\mathrm{y}=\beta ,\mathrm{z}=\gamma$. Then the distance of the point ( $\alpha ,\beta ,\gamma$ ) from the plane …

If the system of linear equations
$7x+11y+\alpha z=13$
$5x+4y+7z=\beta$
$175x+194y+57z=361$
has infinitely many solutions, then $\alpha +\beta +2$ is equal to

Let A be a $2\times 2$ matrix with real entries such that ${\mathrm{A}}^{\prime }=\alpha \mathrm{A}+\mathrm{I}$, where $\alpha \in \mathbb{R}-\{-1,1\}$. If $\det (A^2-A)=4$, then the sum of all possible values of $\alpha$ is equal to

Let $\mathrm{A}=\left[\begin{array}{lll}0& 1& 2\\ \mathrm{a}& 0& 3\\ 1& \mathrm{c}& 0\end{array}\right]$, where $\mathrm{a},\mathrm{c}\in \mathrm{R}$. If ${\mathrm{A}}^3=\mathrm{A}$ and the positive value of a belongs to the interval $(\mathrm{n}-1,\mathrm{n}]$ …