Matrices — JEE Main PYQs

The total number of $3\times 3$ matrices A having entries from the set $\{0,1,2,3\}$ such that the sum of all the diagonal entries of $A{A}^{\top }$ is 9, is equal to

Let $\mathrm{P}=\left[\begin{array}{ccc}-30& 20& 56\\ 90& 140& 112\\ 120& 60& 14\end{array}\right]$ and $\mathrm{A}=\left[\begin{array}{ccc}2& 7& {\omega }^2\\ -1& -\omega & 1\\ 0& -\omega & -\omega +1\end{array}\right]$ where $\omega =\frac{-1+\mathrm{i}\sqrt{3}}{2}$ …

Let $A=\left[\begin{array}{l}{a}_1\\ a_2\end{array}\right]$ and $B=\left[\begin{array}{l}{b}_1\\ b_2\end{array}\right]$ be two $2\times 1$ matrices with real entries such that $A=XB$, where …

Let the system of linear equations
$4x+\lambda y+2z=0$
$2x-y+z=0$
$\mu x+2y+3z=0,\lambda ,\mu \in R$
has a non-trivial solution. Then which of the following is true?

Let I be an identity matrix of order $2\times 2$ and $\mathrm{P}=\left[\begin{array}{cc}2& -1\\ 5& -3\end{array}\right]$. Then the value of $\mathrm{n}\in \mathrm{N}$ for which $P^n=5I-8P$ is equal to

Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{l}\alpha \\ \beta \end{array}\right]\ne \left[\begin{array}{l}0\\ 0\end{array}\right]$ such that $\mathrm{AB}=\mathrm{B}$ and …

Let $A+2B=\left[\begin{array}{ccc}1& 2& 0\\ 6& -3& 3\\ -5& 3& 1\end{array}\right]$ and $2A-B=\left[\begin{array}{ccc}2& -1& 5\\ 2& -1& 6\\ 0& 1& 2\end{array}\right]$. If $\mathrm{Tr}(A)$ denotes the sum of all diagonal elements of the matrix $A$, t …

If the system of linear equations,

has more than two solutions, then $\mu -{\lambda }^2$ is equal to ____

For which of the following ordered pairs $(\mu ,\delta )$, the system of linear equations
$x+2y+3z=1$
$3x+4y+5z=\mu$
$4x+4y+4z=\delta$
is inconsistent?

The number of all $3\times 3$ matrices, $A$, with entries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $A{A}^{\top }$ is 3, is ____.

The system of linear equations
$\lambda x+2y+2z=5$
$2\lambda x+3y+5z=8$
$4x+\lambda y+6z=10$ has:

If $A=\left(\begin{array}{cc}2& 2\\ 9& 4\end{array}\right)$ and $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, then $10A^{-1}$ is equal to:

The following system of linear equation
$7x+6y-2z=0$
$3x+4y+2z=0$ has
$x-2y-6z=0$,

If the matrices $\mathrm{A}=\left[\begin{array}{ccc}1& 1& 2\\ 1& 3& 4\\ 1& -1& 3\end{array}\right],\mathrm{B}=\mathrm{adj}\mathrm{A}$ and $\mathrm{C}=3\mathrm{A}$, then $\frac{|\mathrm{adj}\mathrm{B}|}{|\mathrm{C}|}$ is equal to:

If for some $\alpha$ and $\beta$ in R, the intersection of the following three planes
$x+4y-2z=1$
$x+7y-5z=\beta$
$x+5y+\alpha z=5$
is a line in ${\mathrm{R}}^3$, then $\alpha +\beta$ is equal to:

If the system of linear equations
$x+y+3z=0$
$\mathrm{x}+3\mathrm{y}+{\mathrm{k}}^2\mathrm{z}=0$
$3x+y+3z=0$
has a non-zero solution $(x,y,z)$ for some $k\in R$, then $x+\left(\frac{y}{z}\right)$ is equal to:

Let $A=\left[\begin{array}{ll}x& 1\\ 1& 0\end{array}\right],x\in R$ and $A^4=\left[a_{ij}\right]$. If $a_{11}=109$, then $a_{22}$ is equal to ______.

Let $A$ be a $3\times 3$ matrix such that $\mathrm{adj}A=\left[\begin{array}{ccc}2& -1& 1\\ -1& 0& 2\\ 1& -2& -1\end{array}\right]$ and $B=\mathrm{adj}(\mathrm{adj}A)$. If $|A|=\lambda$ and $\left|{\left(B^{-1}\right)}^{\top }\right|=\mu$, th …

Let S be the set of all integer solutions, $(\mathrm{x},\mathrm{y},\mathrm{z})$, of the system of equation $x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0$
such that $15\le x^2+y^2+z^2\le 150$. Then, the number of elements in the set $S$ is equal to ______.

Let $\theta =\frac{\pi }{5}$ and $\mathrm{A}=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$. If $\mathrm{B}=\mathrm{A}+{\mathrm{A}}^4$, then $\det (\mathrm{B})$ :

Let $\lambda \epsilon \mathrm{R}$. The system of linear equations.
$2x_1-4x_2+\lambda x_3=1$
$x_1-6x_2+x_3=2$
$\lambda {\mathrm{x}}_1-10{\mathrm{x}}_2+4{\mathrm{x}}_3=3$
Is inconsistent for:

The value of $\lambda$ and $\mu$ for which the system of linear equations

has infinitely many solutions are, respectively:

If $A=\left[\begin{array}{cc}\cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{array}\right],\left(\theta =\frac{\pi }{24}\right)$ and $A^5=\left[\begin{array}{ll}a& b\\ c& d\end{array}\right]$, where $i=\sqrt{-1}$, then, which one of the following is not tr …

If the system of equations
$x-2y+3z=9$
$2\mathrm{x}+\mathrm{y}+\mathrm{z}=\mathrm{b}$
$x-7y+az=24$,
has infinitely many solutions, then $\mathrm{a}-\mathrm{b}$ is equal to ____ .

If the system of equations
$x+y+z=2$
$2x+4y-z=6$
$3x+2y+\lambda z=\mu$
has infinitely many solutions, then:

Suppose the vectors $x_1$, $x_2$ and $x_3$ are the solutions of the system of linear equations, $Ax=b$ when the vector $b$ on the right side is equal to $b_1$, $b_2$ and $b_3$ respectively. If $x_1=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$, …

Let $\mathrm{A}=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right),(\alpha \in \mathrm{R})$ such that ${\mathrm{A}}^{32}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. Then a value of $\alpha$ is:

The greatest value of $\mathrm{c}\in \mathrm{R}$ for which the system of linear equations
$x-cy-cz=0$
$cx-y+cz=0$
$cx+cy-z=0$
has a non - trivial solution, is:

If the system of linear equations
$\mathrm{x}-2\mathrm{y}+\mathrm{kz}=1$
$2\mathrm{x}+\mathrm{y}+\mathrm{z}=2$
$3x-y-kz=3$
has a solution $(x,y,z),z\ne 0$, then $(x,y)$ lies on the straight line whose equation is:

If …

If the system of equations

has infinitely many solutions, then $\beta -\alpha$ equals:

Let $A$ and $B$ be two invertible matrices of order $3\times 3$. If $\det \left(AB{A}^{\top }\right)=8$ and $\det \left(A{B}^{-1}\right)=8$, then $\det \left(B{A}^{-1}{B}^{\top }\right)$ is equal to:

If $B=\left[\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right]$ is the inverse of a $3\times 3$ matrix $A$, then the sum of all values of $\alpha$ for which $\det (A)+1=0$, is :

If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that $A+B=\left[\begin{array}{cc}2& 3\\ 5& -1\end{array}\right]$, then $AB$ is equal to :

Let $\lambda$ be a real number for which the system of linear equations
$x+y+z=6$
$4x+\lambda y-\lambda z=\lambda -2$
$3x+2y-4z=-5$
Has indefinitely many solutions. Then $\lambda$ is a root of the quadratic equation

If the system of linear equations
$2x+2y+3z=a$
$3x-y+5z=b$
$x-3y+2z=c$
Where $\mathrm{a},\mathrm{b},\mathrm{c}$ are non zero real numbers, has more than one solution, then:

Let $A=\left(\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right)$. If $A{A}^{\top }=I_3$, then $|p|$ is:

An ordered pair ($\alpha ,\beta$) for which the system of linear equations
$(1+\alpha )x+\beta y+z=2$
$\alpha x+(1+\beta )y+z=3$
$\alpha x+\beta y+2z=2$
has a unique solution, is:

Let $P=\left[\begin{array}{lll}1& 0& 0\\ 3& 1& 0\\ 9& 3& 1\end{array}\right]$ and $Q=\left[q_{ij}\right]$ be two $3\times 3$ matrices such that $Q-P^5=I_3$. Then $\frac{q_{21}+q_{31}}{q_{32}}$ is equal to:

The set of all values of $\lambda$ for which the system of linear equations
$x-2y-2z=\lambda x$
$x+2y+z=\lambda y$
$-x-y=\lambda z$

If the system of equations $2x+3y-z=0,x+ky-2z=0$ and $2x-y+z=0$ has a non trivial solution $(x,y,z)$, then $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+k$ is equal to

The total number of matrices $A=\left[\begin{array}{ccc}0& 2x& 2x\\ 2y& y& -y\\ 1& -1& 1\end{array}\right]$; $(x,y\in R,x\ne y)$ for which $A^{\top }A=3I_3$

If the system of linear equation $x-4y+7z=g,3y-5z=h,-2x+5y-9z=k$ is consistent, then:

If $\mathrm{A}=\left[\begin{array}{ccc}{e}^t& e^{-t}\cos t& e^{-t}\sin t\\ e^t& -e^{-t}\cos t-e^{-t}\sin t& -e^{-t}\sin t+e^{-t}\cos t\\ e^t& 2e^{-t}\sin t& -2e^{-t}\cos t\end{array}\right]$ Then $A$ is

The system of linear equation $x+y+z=2,2x+3y+2z=5,2x+3y+\left(a^2-1\right)z=a+1$ then

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then the matrix $A^{-50}$ when $\theta =\frac{\pi }{12}$, is equal to

If the system of linear equations

$x+3y+\lambda z=\mu$ ($\lambda ,\mu \in R$), has infinitely many solutions then the value of $\lambda +\mu$ is:

If the system of linear equations

has a non-zero solution $(x,y,z)$, then $\frac{xz}{y^2}$ is equal to :

If S is the set of distinct values of ' b ' for which the following system of linear equations

has no solution, then S is :

If $A=\left[\begin{array}{rr}2& -3\\ -4& 1\end{array}\right]$, then $\mathrm{adj}\left(3A^2+12A\right)$ is equal to :