Matrices — JEE Main PYQs

For some $\alpha ,\beta \in R$, let $A=\left[\begin{array}{ll}\alpha & 2\\ 1& 2\end{array}\right]$ and $B=\left[\begin{array}{ll}1& 1\\ 1& \beta \end{array}\right]$ be such that $A^2-4A+2I=B^2-3B+I=O$. Then …

For the matrices $\mathrm{A}=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}-29& 49\\ -13& 18\end{array}\right]$, if …

If the system of equations
$3\mathrm{x}+\mathrm{y}+4\mathrm{z}=3$
$2\mathrm{x}+\alpha \mathrm{y}-\mathrm{z}=-3$
$\mathrm{x}+2\mathrm{y}+\mathrm{z}=4$
has no solution, then the value of $\alpha$ is equal to:

If $\mathrm{A}=\left[\begin{array}{ll}2& 3\\ 3& 5\end{array}\right]$, then the determinant of the matrix $({\mathrm{A}}^{2025}-3{\mathrm{A}}^{2024}+{\mathrm{A}}^{2023})$ is

Let A be a $3\times 3$ matrix such that $\mathrm{A}+{\mathrm{A}}^{\mathrm{T}}=\mathrm{O}$. If …

If $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ is a solution of the system of equations $AX=B$, where $\text{adj }A=\left[\begin{array}{ccc}4& 2& 2\\ -5& 0& 5\\ 1& -2& 3\end{array}\right]$ and $B=\left[\begin{array}{c}4\\ 0\\ 2\end{array}\right]$, then $|x+y+z|$ is equal t …

Let $|A|=6$, where A is a $3\times 3$ matrix. If $|\mathrm{adj}(3\mathrm{adj}(A^2.\mathrm{adj}(2A)))|=2^m\cdot 3^n,m,n\in \mathbf{N}$, then $m+n$ is equal to ______.

The system of linear equations
$x+y+z=6$
$2\mathrm{x}+5\mathrm{y}+\mathrm{az}=36$
$x+2y+3z=b$
has

Let $\mathrm{A}=\left[\begin{array}{ccc}0& 2& -3\\ -2& 0& 1\\ 3& -1& 0\end{array}\right]$ and B be a matrix such that $\mathrm{B}(\mathrm{I}-\mathrm{A})=\mathrm{I}+\mathrm{A}$. Then the sum of the diagonal elements of $B^{T}B$ is equal to

The number of $3\times 2$ matrices A, which can be formed using the elements of the set $\{-2,-1,0,1,2\}$ such that the sum of all the diagonal elements of $A^{T}A$ is 5, is ______

Let $\mathrm{P}=[{\mathrm{p}}_{\mathrm{ij}}]$ and $\mathrm{Q}=[{\mathrm{q}}_{\mathrm{ij}}]$ be two square matrices of order 3 such that ${\mathrm{q}}_{\mathrm{ij}}=2^{(\mathrm{i}+\mathrm{j}-1)}{\mathrm{p}}_{\mathrm{ij}}$ and $\det (\mathrm{Q})=2^{10}$. Then the val …

Let $\mathrm{A},\mathrm{B}$ and C be three $2\times 2$ matrices with real entries such that $\mathrm{B}=(\mathrm{I}+\mathrm{A}{)}^{-1}$ and $\mathrm{A}+\mathrm{C}=\mathrm{I}$. If $\mathrm{BC}=\left[\begin{array}{cc}1& -5\\ -1& 2\end{array}\right]$ and …

Let $\mathrm{A}=\left[\begin{array}{ll}3& -4\\ 1& -1\end{array}\right]$ and B be two matrices such that $A^{100}=100B+I$. Then the sum of all the elements of ${\mathrm{B}}^{100}$ is ______ .

Let $\alpha ,\beta \in \mathbf{R}$ be such that the system of linear equations
$x+2y+z=5$
$2x+y+\alpha z=5$
$8x+4y+\beta z=18$
has no solution. Then $\frac{\beta }{\alpha }$ is equal to :

Let $A=\left[\begin{array}{cc}1& 2\\ 1& \alpha \end{array}\right]$ and $B=\left[\begin{array}{cc}3& 3\\ \beta & 2\end{array}\right]$.

If $A^2-4A+I=O$ and $B^2-5B-6I=O$, then among the two statements :
(S1) : $[(B-A)(B+A){]}^T=\left[\begin{array}{cc}13& 15\\ 7& 10\end{array}\right]$
and
(S2): …

If the system of equations :
$x+y+z=5$
$x+2y+3z=9$
$x+3y+\lambda z=\mu$
has infinitely many solutions, then the value of $\lambda +\mu$ is :

Let $\mathrm{A}=\left[\begin{array}{ccc}1& 2& 7\\ 4& -2& 8\\ 3& 8& -7\end{array}\right]$ and $\det (\mathrm{A}-\alpha \mathrm{I})=0$, where $\alpha$ is a real number. If the largest possible value of $\alpha$ is $p$, then the circle …

Let $\mathrm{S}=\{\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\in \{0,1,2,3,4\}$ and ${\mathrm{A}}^2-4\mathrm{A}+3\mathrm{I}=0\}$ be a set of …

Let $\mathrm{A}=\left[\begin{array}{ccc}1& 1& 2\\ -2& 0& 1\\ 1& 3& 5\end{array}\right]$. Then the sum of all elements of the matrix $\mathrm{adj}\left(\mathrm{adj}\left(2(\mathrm{adj}\mathrm{A}{)}^{-1}\right)\right)$ is equal to :

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

Consider the system of linear equations in $\mathrm{x},\mathrm{y},\mathrm{z}$ :
$\mathrm{x}+2\mathrm{y}+\mathrm{tz}=0$,
$6\mathrm{x}+\mathrm{y}+5\mathrm{tz}=0$,
$3\mathrm{x}+{\mathrm{t}}^2\mathrm{y}+\mathrm{f}(\mathrm{t})\mathrm{z}=0$,
Where …

Let M be a $3\times 3$ matrix such that
$\mathrm{M}\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right),\mathrm{M}\left(\begin{array}{l}0\\ 1\\ 0\end{array}\right)=\left(\begin{array}{l}0\\ 1\\ 2\end{array}\right)$ and …

Let $\mathrm{A}=\left[\begin{array}{lll}1& 0& 0\\ 3& 1& 0\\ 9& 3& 1\end{array}\right]$ and $\mathrm{B}=\left[{\mathrm{b}}_{\mathrm{ij}}\right],1\le \mathrm{i},\mathrm{j}\le 3$.

If $B=A^{99}-I$, then the value of $\frac{b_{31}-b_{21}}{b_{32}}$ is:

Let $\mathrm{A}=\left[\begin{array}{ccc}-1& 1& -1\\ 1& 0& 1\\ 0& 0& 1\end{array}\right]$ satisfy
${\mathrm{A}}^2+\alpha (\mathrm{adj}(\mathrm{adj}(\mathrm{A})))+\beta (\mathrm{adj}(\mathrm{A})(\mathrm{adj}(\mathrm{adj}(\mathrm{A}))))$ …

If the system of linear equations:
$x+y+z=6$,
$x+2y+5z=10$,
$2\mathrm{x}+3\mathrm{y}+\lambda \mathrm{z}=\mu$.
has infinitely many solutions, then the value of $\lambda +\mu$ equals.

Let $A=\left[\begin{array}{ccc}\alpha & 1& 2\\ 2& 3& 0\\ 0& 4& 5\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}1& 0& 0\\ 0& -5\alpha & 0\\ 0& 4\alpha & -2\alpha \end{array}\right]+\mathrm{adj}(\mathrm{A})$. If …

If the system of equations

has infinitely many solutions, then the point $(\mathrm{a},\mathrm{b})$ lies on the line

Consider the matrices $\mathrm{A}=\left[\begin{array}{cc}2& -2\\ 4& -2\end{array}\right]$ and $\mathrm{B}\left[\begin{array}{ll}3& 9\\ 1& 3\end{array}\right]$. If matrices P and Q are such that $\mathrm{PA}=$ B and $\mathrm{AQ}=\mathrm{B}$, then the absolute value o …

Let A be a square matrix of order 3 such that $\det (\mathrm{A})=-2$ and $\det (3\mathrm{adj}(-6\mathrm{adj}(3\mathrm{A})))=2^{\mathrm{m}+\mathrm{n}}\cdot 3^{\mathrm{mn}}$, where $\mathrm{m}>\mathrm{n}$. Then $4\mathrm{m}+2\mathrm{n}$ i …

For a $3\times 3$ matrix M , let trace $(\mathrm{M})$ denote the sum of all the diagonal elements of M . Let A be a $3\times 3$ matrix such that $|\mathrm{A}|=\frac{1}{2}$ and trace $(\mathrm{A})=3$. If $B=\mathrm{adj}(\mathrm{adj}(2A))$, then the value of $|B|+$ t …

If the system of linear equations :
$x+y+2z=6$,
$2\mathrm{x}+3\mathrm{y}+\mathrm{az}=\mathrm{a}+1$,
$-\mathrm{x}-3\mathrm{y}+\mathrm{bz}=2\mathrm{b}$,
where $a,b\in \mathbf{R}$, has infinitely many solutions, then $7\mathrm{a}+3\mathrm{b}$ is equal to :

If $\mathrm{A},\mathrm{B}$ and $\left(\mathrm{adj}\left({\mathrm{A}}^{-1}\right)+\mathrm{adj}\left({\mathrm{B}}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of …

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

The system of equations
$\mathrm{x}+\mathrm{y}+\mathrm{z}=6$,
$x+2y+5z=9$,
$\mathrm{x}+5\mathrm{y}+\lambda \mathrm{z}=\mu$,
has no solution if

Let $A=[a_{ij}]$ be a $3\times 3$ matrix such that
$A\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{l}0\\ 0\\ 1\end{array}\right],A\left[\begin{array}{l}4\\ 1\\ 3\end{array}\right]=\left[\begin{array}{l}0\\ 1\\ 0\end{array}\right]$ and …

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a matrix of order $3\times 3$, with ${\mathrm{a}}_{\mathrm{ij}}=(\sqrt{2}{)}^{\mathrm{i}+\mathrm{j}}$. If the sum of all the elements in the third row of ${\mathrm{A}}^2$ is …

Let $S=\left\{m\in Z:A^{m^2}+A^m=3I-A^{-6}\right\}$, where $\mathrm{A}=\left[\begin{array}{cc}2& -1\\ 1& 0\end{array}\right]$. Then $\mathrm{n}(S)$ is equal to ______.

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}}& -2\\ 0& 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right],\theta >0$. If …

Let M denote the set of all real matrices of order $3\times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let
$S_1=\left\{A=\left[a_{ij}\right]\in M:A=A^T\text{ and }{a}_{ij}\in S,\forall i,j\right\}$ …

Let be a $3\times 3$ matrix such that ${\mathrm{X}}^{\mathrm{T}}\mathrm{AX}=\mathrm{O}$ for all nonzero $3\times 1$ matrices $\mathrm{X}=\left[\begin{array}{l}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$.

If …

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

Let A be a $3\times 3$ real matrix such that $A^2(A-2I)-4(A-I)=O$, where I and O are the identity and null matrices, respectively. If $A^5=\alpha A^2+\beta A+\gamma I$, where $\alpha ,\beta$ and $\gamma$ are real constants, then $\alpha +\beta +\gamma$ is equal to:

Let I be the identity matrix of order $3\times 3$ and for the matrix $A=\left[\begin{array}{ccc}\lambda & 2& 3\\ 4& 5& 6\\ 7& -1& 2\end{array}\right],|A|=-1$. Let $B$ be the inverse of the matrix …

Let $A=\left[\begin{array}{cc}\alpha & -1\\ 6& \beta \end{array}\right]$, $\alpha >0$, such that $\det (A)=0$ and $\alpha +\beta =1$. If $I$ denotes $2\times 2$ identity matrix, then the matrix $(I+A{)}^8$ is:

If the system of linear equations
$3x+y+\beta z=3$
$2\mathrm{x}+\alpha \mathrm{y}-\mathrm{z}=-3$
$\mathrm{x}+2\mathrm{y}+\mathrm{z}=4$
has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

Let $\mathrm{a}\in \mathbf{R}$ and A be a matrix of order $3\times 3$ such that $\det (\mathrm{A})=-4$ and $\mathrm{A}+\mathrm{I}=\left[\begin{array}{lll}1& \mathrm{a}& 1\\ 2& 1& 0\\ \mathrm{a}& 1& 2\end{array}\right]$, where I is the identity matri …

Let A be a matrix of order $3\times 3$ and $|\mathrm{A}|=5$. If $|2\mathrm{adj}(3\mathrm{A}\mathrm{adj}(2\mathrm{A}))|=2^{\alpha }\cdot 3^{\beta }\cdot 5^{\gamma }$ where $\alpha ,\beta ,\gamma \in \mathrm{N}$, then $\alpha +\beta +\gamma$ is equal to

Let the matrix $A=\left[\begin{array}{lll}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $\mathrm{n}\ge 3$. Then the sum of all the elements of ${\mathrm{A}}^{50}$ is :-

Let $\mathrm{A}=\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]$. If for some $\theta \in (0,\pi )$, $A^2=A^T$, then the sum of the diagonal elements of the matrix …

Let the system of equations
$\mathrm{x}+5\mathrm{y}-\mathrm{z}=1$
$4\mathrm{x}+3\mathrm{y}-3\mathrm{z}=7$
$24\mathrm{x}+\mathrm{y}+\lambda \mathrm{z}=\mu$
$\lambda ,\mu \in \mathrm{R}$, have infinitely many solutions. Then the number of the solutions of this system, …