Matrices — JEE Main PYQs

If A and B are two non-zero $\mathrm{n}\times \mathrm{n}$ matrics such that ${\mathrm{A}}^2+\mathrm{B}={\mathrm{A}}^2\mathrm{B}$, then

Let $A=\left(\begin{array}{cc}m& n\\ p& q\end{array}\right),d=|A|\ne 0,|A-d(\mathrm{Adj}A)|=0$. Then

Let the system of linear equations
$x+y+kz=2$
$2x+3y-z=1$
$3x+4y+2z=k$
have infinitely many solutions. Then the system
$(k+1)x+(2k-1)y=7$
$(2k+1)x+(k+5)y=10$ has:

If $\mathrm{A}=\frac{1}{2}\left[\begin{array}{cc}1& \sqrt{3}\\ -\sqrt{3}& 1\end{array}\right]$, then :

For the system of linear equations $\mathrm{ax}+\mathrm{y}+\mathrm{z}=1$, $x+ay+z=1,x+y+az=\beta$, which one of the following statements is NOT correct ?

The set of all values of $t\in \mathbb{R}$, for which the matrix

i …

Let A be a symmetric matrix such that $|\mathrm{A}|=2$ and $\left[\begin{array}{ll}2& 1\\ 3& \frac{3}{2}\end{array}\right]\mathrm{A}=\left[\begin{array}{ll}1& 2\\ \alpha & \beta \end{array}\right]$. If the sum of the diagonal elements of A is s, then …

Consider the following system of equations
$\alpha \mathrm{x}+2\mathrm{y}+\mathrm{z}=1$
$2\alpha \mathrm{x}+3\mathrm{y}+\mathrm{z}=1$
$3\mathrm{x}+\alpha \mathrm{y}+2\mathrm{z}=\beta$
For some $\alpha ,\beta \in \mathbb{R}$. Then which of the following is NOT correct …

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix A such that ${\mathrm{A}}^2=3\mathrm{A}+\alpha \mathrm{I}$. If ${\mathrm{A}}^4=21\mathrm{A}+\beta \mathrm{I}$, then

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{10}}& \frac{3}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}& \frac{1}{\sqrt{10}}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}1& -\mathrm{i}\\ 0& 1\end{array}\right]$, where $\mathrm{i}=\sqrt{-1}$. If …

Let $\mathrm{A},\mathrm{B},\mathrm{C}$ be $3\times 3$ matrices such that A is symmetric and B and C are skew-symmetric.

Consider the statements
(S1) ${\mathrm{A}}^{13}{\mathrm{B}}^{26}-{\mathrm{B}}^{26}{\mathrm{A}}^{13}$ is symmetric
(S2) …

If the system of equations
$x+2y+3z=3$
$4x+3y-4z=4$
$8x+4y-\lambda z=9+\mu$
has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

Let A be a $3\times 3$ matrix such that $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))|=12^4$. Then $\left|A^{-1}\mathrm{adj}A\right|$ is equal to

If a point $\mathrm{P}(\alpha ,\beta ,\gamma )$ satisfying $(\alpha \beta \gamma )\left(\begin{array}{ccc}2& 10& 8\\ 9& 3& 8\\ 8& 4& 8\end{array}\right)=\left(\begin{array}{lll}0& 0& 0\end{array}\right)$ lies on the plane $2\mathrm{x}+4\mathrm{y}+3\mathrm{z}=5$, …

Let A be a $\mathrm{n}\times \mathrm{n}$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix $\mathrm{Adj}\left(2.\mathrm{Adj}\left(2{\mathrm{A}}^{-1}\right)\right)$ is $2^{84}$, then n is equal to ______.

If P is a $3\times 3$ real matrix such that ${\mathrm{P}}^{\mathrm{T}}=\mathrm{aP}+(\mathrm{a}-1)\mathrm{I}$, where $\mathrm{a}>1$, then

For $\alpha ,\beta \in \mathbb{R}$, suppose the system of linear equations
$\mathbf{x}-y+z=5$
$2\mathrm{x}+2\mathrm{y}+\alpha \mathrm{z}=8$
$3x-y+4z=\beta$
has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of

For the system of linear equations
$x+y+z=6$
$\alpha x+\beta y+7z=3$
$x+2y+3z=14$,
which of the following is NOT true?

Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 0& 4& -1\\ 0& 12& -3\end{array}\right)$. Then the sum of the diagonal elements of the matrix $(A+I{)}^{11}$ is equal to:

Let S denote the set of all real values of $\lambda$ such that the system of equations
$\lambda \mathrm{x}+\mathrm{y}+\mathrm{z}=1$
$\mathrm{x}+\lambda \mathrm{y}+\mathrm{z}=1$
$\mathrm{x}+\mathrm{y}+\lambda \mathrm{z}=1$
is inconsistent, then …

Let ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ be respectively the sets of all $\mathrm{a}\in \mathrm{R}-\{0\}$ for which the system of linear equations
$ax+2ay-3az=1$
$(2a+1)x+(2a+3)y+(a+1)z=2$
$(3a+5)x+(a+5)y+(a+2)z=3$
has unique solution and infinitely ma …

Which of the following matrices can NOT be obtained from the matrix $\left[\begin{array}{cc}-1& 2\\ 1& -1\end{array}\right]$ by a single elementary row operation?

If the system of equations
$x+y+z=6$
$2x+5y+\alpha z=\beta$
$x+2y+3z=14$
has infinitely many solutions, then $\alpha +\beta$ is equal to:

Let $\mathrm{X}=\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ccc}-1& 2& 3\\ 0& 1& 6\\ 0& 0& -1\end{array}\right]$. For $\mathrm{k}\in \mathbb{N}$, if ${\mathrm{X}}^{\prime }{\mathrm{A}}^k\mathrm{X}=33$, then k is equal to:

Let the matrix $\mathrm{A}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\mathrm{Adj}\left(B_{n-1}\right)$ for all $n\ge 1$, then $\det \left({\mathrm{B}}_4\right)$ is equal to …

Let $\mathrm{A}=\left[\begin{array}{cc}1& -1\\ 2& \alpha \end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{cc}\beta & 1\\ 1& 0\end{array}\right],\alpha ,\beta \in \mathrm{R}$. Let ${\alpha }_1$ be the value of $\alpha$ which satisfies …

Let A and B be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let $A=\left(\begin{array}{cc}1& 2\\ -2& -5\end{array}\right)$. Let $\alpha ,\beta \in \mathbb{R}$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha +\beta$ is equal to -

Let S be the set containing all $3\times 3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is 6 is ______.

Let $A=\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\\ 1& -1& 0\end{array}\right]$ and $B=A-I$. If $\omega =\frac{\sqrt{3}i-1}{2}$, then the number of elements in the set $\{n\in \{1,2,\dots ,100\}:A^n+(\omega B{)}^n=A+B\}$ is equal to ____.

Let $A=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}{9}^2& -10^2& 11^2\\ 12^2& 13^2& -14^2\\ -15^2& 16^2& 17^2\end{array}\right]$, then the value of $A^{\prime }BA$ is:

The number of matrices $A=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$, where $\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\in \{-1,0,1,2,3,\dots ,10\}$, such that $\mathrm{A}={\mathrm{A}}^{-1}$, is ______.

Let $\mathrm{A}=\left(\begin{array}{cc}4& -2\\ \alpha & \beta \end{array}\right)$

If ${\mathrm{A}}^2+\gamma \mathrm{A}+18\mathrm{I}=\mathrm{O}$, then $\det (\mathrm{A})$ is equal to
____ .

The number of real values $\lambda$, such that the system of linear equations
$2\mathrm{x}-3\mathrm{y}+5\mathrm{z}=9$
$\mathrm{x}+3\mathrm{y}-\mathrm{z}=-18$
$3x-y+\left({\lambda }^2-|\lambda |\right)z=16$
has no solution, is :-

Let $\mathrm{A}=\left[\begin{array}{ccc}1& \mathrm{a}& \mathrm{a}\\ 0& 1& \mathrm{b}\\ 0& 0& 1\end{array}\right],\mathrm{a},\mathrm{b}\in \mathbb{R}$. If for some $\mathrm{n}\in \mathrm{N}$, …

If the system of linear equations
$8x+y+4z=-2$
$x+y+z=0$
$\lambda \mathrm{x}-3\mathrm{y}=\mu$
has infinitely many solutions, then the distance of the point $\left(\lambda ,\mu ,-\frac{1}{2}\right)$ from the plane $8x+y+4z+2=0$ is :

Let A be a $2\times 2$ matrix with $\det (\mathrm{A})=-1$ and $\det ((\mathrm{A}+\mathrm{I})(\mathrm{Adj}(\mathrm{A})+\mathrm{I}))=4$. Then the sum of the diagonal elements of A can be :

If the system of linear equations
$2\mathrm{x}-3\mathrm{y}=\gamma +5$,
$\alpha \mathrm{x}+5\mathrm{y}=\beta +1$, where $\alpha ,\beta ,\gamma \in \mathbf{R}$ has infinitely many solutions, then the value of $|9\alpha +3\beta +5\gamma |$ is equal to

Let $\mathrm{A}=\left(\begin{array}{cc}1+\mathrm{i}& 1\\ -\mathrm{i}& 0\end{array}\right)$ where $\mathrm{i}=\sqrt{-1}$.

Then, the number of elements in the set
$\left\{\mathrm{n}\in \{1,2,\dots ,100\}:{\mathrm{A}}^{\mathrm{n}}=\mathrm{A}\right\}$ is

Let A and B be two $3\times 3$ matrices such that $AB=I$ and $|A|=\frac{1}{8}$ then $|\mathrm{adj}(B\mathrm{adj}(2A))|$ is equal to

If the system of linear equations
$2x+3y-z=-2$
$x+y+z=4$
$\mathrm{x}-\mathrm{y}+|\lambda |\mathrm{z}=4\lambda -4$
where $\lambda \in \mathbb{R}$, has no solution, then

Let A be a matrix of order $3\times 3$ and $\det (\mathrm{A})=2$. Then $\det \left(\det (\mathrm{A})\mathrm{adj}\left(5\mathrm{adj}\left({\mathrm{A}}^3\right)\right)\right)$ is equal to

Let $A=\left(\begin{array}{cc}2& -1\\ 0& 2\end{array}\right)$. If $B=I-{}^5{C}_1(\text{adj}A)+{}^5{C}_2(\text{adj}A{)}^2-\dots -{}^5{C}_5(\text{adj}A{)}^5$, then the sum of all elements of the matrix $B$ is:

Let $\mathrm{M}=\left[\begin{array}{cc}0& -\alpha \\ \alpha & 0\end{array}\right]$, where $\alpha$ is a non-zero real number and $N={\sum }_{k=1}^{49}{M}^{2k}$. If $\left(I-M^2\right)N=-2I$, then the positive integral value of $\alpha$ is ______.

If the system of equations $\alpha x+y+z=5,x+2y+3\mathrm{z}=4,\mathrm{x}+3\mathrm{y}+5\mathrm{z}=\beta$,
has infinitely many solutions, then the ordered pair $(\alpha ,\beta )$ is equal to :

Let $X=\left[\begin{array}{lll}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right],Y=\alpha I+\beta X+\gamma X^2$ and $\mathrm{Z}={\alpha }^2\mathrm{I}-\alpha \beta \mathrm{X}+\left({\beta }^2-\alpha \gamma \right){\mathrm{X}}^2,\alpha ,\beta ,\gamma \in \mathbb{R}$. If …

If the system of linear equations
$2\mathrm{x}+\mathrm{y}-\mathrm{z}=7$
$\mathrm{x}-3\mathrm{y}+2\mathrm{z}=1$
$\mathrm{x}+4\mathrm{y}+\delta \mathrm{z}=\mathrm{k}$, where $\delta ,\mathrm{k}\in \mathrm{R}$
has infinitely many solutions, then $\delta +\mathrm{k}$ is …

Let $\mathrm{A}=\left[{\mathrm{a}}_{\mathrm{ij}}\right]$ be a square matrix of order 3 such that ${\mathrm{a}}_{\mathrm{ij}}=2^{\mathrm{j}-\mathrm{i}}$, for all $\mathrm{i},\mathrm{j}=1,2,3$. Then, the matrix ${\mathrm{A}}^2+{\mathrm{A}}^3+\dots +{\mathrm{A}}^{10}$ is equal to …

Let A be a $3\times 3$ invertible matrix. If $|\mathrm{adj}(24\mathrm{A})|=|\mathrm{adj}(3\mathrm{adj}(2\mathrm{A}))|$, then $|\mathrm{A}{|}^2$ is equal to :

The ordered pair (a, b), for which the system of linear equations
$3\mathrm{x}-2\mathrm{y}+\mathrm{z}=\mathrm{b}$
$5\mathrm{x}-8\mathrm{y}+9\mathrm{z}=3$
$2\mathrm{x}+\mathrm{y}+\mathrm{az}=-1$
has no solution, is :