Matrices — JEE Main PYQs

Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha$, $-\alpha x+y+2z=-\alpha$ be inconsistent.

Then $\alpha$ is equal to :

The positive value of the determinant of the matrix A, whose $\mathrm{Adj}(\mathrm{Adj}(A))=\left(\begin{array}{ccc}14& 28& -14\\ -14& 14& 28\\ 28& -14& 14\end{array}\right)$, is ____.

Let A and B be two $3\times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

The number of matrices of order $3\times 3$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is ______.

The number of values of $\alpha$ for which the system of equations :
$\mathrm{x}+\mathrm{y}+\mathrm{z}=\alpha$
$\alpha \mathrm{x}+2\alpha \mathrm{y}+3\mathrm{z}=-1$
$x+3\alpha y+5z=4$
is inconsistent, is

Let $S=\left\{\left(\begin{array}{cc}-1& a\\ 0& b\end{array}\right);a,b\in \{1,2,3,\dots 100\}\right\}$ and let $T_n=\left\{A\in S:A^{n(n+1)}=I\right\}$. Then the number of elements in ${\bigcap }_{n=1}^{100}{T}_n$ is ______.

The system of equations
$-kx+3y-14z=25$
$-15x+4y-kz=3$
$-4x+y+3z=4$
is consistent for all k in the set

Let $A=\left(\begin{array}{ll}2& -2\\ 1& -1\end{array}\right)$ and $B=\left(\begin{array}{ll}-1& 2\\ -1& 2\end{array}\right)$. Then the number of elements in the set
$\{(n,m):n,m\in \{1,2,\dots ,10\}$ and $n{A}^n+m{B}^m=I\}$
is

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

Let $\mathrm{A}=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$. If M and N are two matrices given by $M={\sum }_{k=1}^{10}{A}^{2k}$ and $N={\sum }_{k=1}^{10}{A}^{2k-1}$ then ${\mathrm{MN}}^2$ is

If $\mathrm{A}=\left(\begin{array}{cc}\frac{1}{\sqrt{5}}& \frac{2}{\sqrt{5}}\\ \frac{-2}{\sqrt{5}}& \frac{1}{\sqrt{5}}\end{array}\right),\mathrm{B}=\left(\begin{array}{cc}1& 0\\ \mathrm{i}& 1\end{array}\right),\mathrm{i}=\sqrt{-1}$, and …

The system of linear equations
$3x-2y-kz=10$
$2\mathrm{x}-4\mathrm{y}-2\mathrm{z}=6$
$x+2y-z=5m$
is inconsistent if

Let M be any $3\times 3$ matrix with entires from the set $\{0,1,2\}$. The maximum number of such matrices, for which the sum of diagonal elements of $M^{\top }M$ is seven is _____.

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$ where $\alpha \in R$. Suppose $Q=[q_{ij}]$ is a matrix satisfying $PQ=k{I}_3$ for some nonzero $k\in R$. If $q_{23}=\frac{-k}{8}$ and $|Q|=\frac{k^2}{2}$, then ${\alpha }^2+k^2$ is equal to ______.

Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations
$x+y+z=5$
$x+2y+3z=\mu$
$x+3y+\lambda z=1$
is constructed. If $\rho$ is the probability that the system has a unique solution $q$ is the probability that …

Let $\mathrm{A}=\left(\begin{array}{lll}1& 0& 0\\ 0& 1& 1\\ 1& 0& 0\end{array}\right)$. Then ${\mathrm{A}}^{2025}-{\mathrm{A}}^{2020}$ is equal to:

Let A be a $3\times 3$ real matrix. If $\det (2\mathrm{adj}(2\mathrm{Adj}(\mathrm{adj}(2\mathrm{A})))=2^{41}$, then the value of $\det \left({\mathrm{A}}^2\right)$ equals ______.

Let $A=\left[\begin{array}{ll}2& 3\\ a& 0\end{array}\right],a\in R$ be written as $P+Q$ where $P$ is a symmetric matrix and $Q$ is skew symmetric matrix. If $\det (Q)=9$, then the modulus of the sum of all possible values of determinant of P is equal t …

Let $A=\left(\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 0& 0& 1\end{array}\right)$ and $B=7A^{20}-20A^7+2I$, where $I$ is an identity matrix of order $3\times 3$. If $B=\left[b_{ij}\right]$, then $b_{13}$ is equal to........

For the system of linear equations:
$x-2y=1,x-y+kz=-2,ky+4z=6,k\in R$,
Consider the following statements:
(A) The system has unique solution if $\mathrm{k}\ne 2,\mathrm{k}\ne -2$.
(B) The system has unique solution if $\mathrm{k}=-2$.
(C) The system has unique solutio …

Let $A$ and $B$ are $3\times 3$ real matrices such that $A$ is symmetric matrix and $B$ is skew- symmetric matrix. Then the system of linear equations $\left(A^2{B}^2-B^2{A}^2\right)X=O$, where $X$ is a $3\times 1$ column matrix of unknown variables and $O$ is a …

The value of $k\in R$, for which the following system of linear equations
$3x-y+4z=3$,
$x+2y-3z=-2$,
$6x+5y+kz=-3$,
has infinitely many solutions, is :

Let $\mathrm{A}=\left\{{\mathrm{a}}_{\mathrm{ij}}\right\}$ be a $3\times 3$ matrix, where …

Let $[\lambda ]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4,3x+2y+5z=3,9x+4y+(28+[\lambda ])z=[\lambda ]$ has a solution is:

If $\mathrm{A}=\left[\begin{array}{cc}0& -\tan \left(\frac{\theta }{2}\right)\\ \tan \left(\frac{\theta }{2}\right)& 0\end{array}\right]$ and …

Let $A=\left[\begin{array}{lll}x& y& z\\ y& z& x\\ z& x& y\end{array}\right]$, where $x,y$ and $z$ are real numbers such that $x+y+z>0$ and $xyz=2$. If $A^2=I_3$, then the value of $x^3+y^3+z^3$ is ____

If the system of equations

has infinitely many solutions, then $k$ is equal to ____

Let $A=|a_{ij}|$ be a real matrix of order $3\times 3$, such that $a_{i1}+a_{i2}+a_{i3}=1$, for $i=1,2,3$. Then, the sum of all the entries of the matrix $A^3$ is equal to:

The values of $\lambda$ and $\mu$ such that the system of equations $x+y+z=6$, $3x+5y+5z=26$, $x+2y+\lambda z=\mu$ has no solution, are:

Let $\mathrm{A}=\left(\begin{array}{lll}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$. Then the number of $3\times 3$ matrices B with entries from the set $\{1,2,3,4,5\}$ and satisfying $AB=BA$ is ____

Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of $A^2$ is 1 , then the possible number of such matrices is :

If the matrix $A=\left(\begin{array}{cc}0& 2\\ K& -1\end{array}\right)$ satisfied $A\left(A^3+3I\right)=2I$, then the value of $K$ is

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

Consider the following system of equations:

where $a,b$ and $c$ are real constants. Then the system of equations :

If the matrix $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 3& 0& -1\end{array}\right]$ satisfies the equation $A^{20}+\alpha A^{19}+\beta A=\left[\begin{array}{lll}1& 0& 0\\ 0& 4& 0\\ 0& 0& 1\end{array}\right]$ for some real numbers $\alpha$ and $\beta$, then …

If $\alpha +\beta +\gamma =2\pi$, then the system of equations
$x+(\cos \gamma )y+(\cos \beta )z=0$
$(\cos \gamma )x+y+(\cos \alpha )z=0$
$(\cos \beta )x+(\cos \alpha )y+z=0$
has:

The number of elements in the set
$\left\{\mathrm{A}=\left(\begin{array}{ll}\mathrm{a}& \mathrm{b}\\ 0& \mathrm{d}\end{array}\right):\mathrm{a},\mathrm{b},\mathrm{d}\in \{-1,0,1\}\mathrm{and}(1-\mathrm{A}{)}^3=\mathrm{I}-{\mathrm{A}}^3\right\}$, where I is …

If the following system of linear equations



has no solution, then:

Let $A=\left[\begin{array}{cc}1& 2\\ -1& 4\end{array}\right]$. If $A^{-1}=\alpha I+\beta A$, $\alpha ,\beta \in R$, $I$ is a $2\times 2$ identity matrix, then $4(\alpha -\beta )$ is equal to :

For real numbers $\alpha$ and $\beta$, consider the following system of linear equations:
$x+y-z=2,x+2y+\alpha z=1,2x-y+z=\beta$.
If the system has infinite solutions, then $\alpha +\beta$ is equal to ____ .

The system of equation $kx+y+z=1$, $x+ky+z=k$ and $x+y+zk=k^2$ has no solution if $k$ is equal to:

If $\mathrm{A}=\left(\begin{array}{cc}0& \sin \alpha \\ \sin \alpha & 0\end{array}\right)$ and $\det \left({\mathrm{A}}^2-\frac{1}{2}\mathrm{I}\right)=0$, then a possible value of $\alpha$ is :

If $\mathrm{A}=\left[\begin{array}{cc}2& 3\\ 0& -1\end{array}\right]$, then the value of $\det \left({\mathrm{A}}^4\right)+\det \left({\mathrm{A}}^{10}-(\mathrm{Adj}(2\mathrm{A}){)}^{10}\right)$ is equal to.....

The values of $a$ and $b$, for which the system of equations
$2x+3y+6z=8$
$x+2y+az=5$
$3\mathrm{x}+5\mathrm{y}+9\mathrm{z}=\mathrm{b}$
Has no solution, are :

Let …

If $\mathrm{P}=\left[\begin{array}{cc}1& 0\\ \frac{1}{2}& 1\end{array}\right]$, then ${\mathrm{P}}^{50}$ is :

Let $A$ and $B$ be two $3\times 3$ real matrices such that $(A^2-B^2)$ is invertible matrix. If $A^5=B^5$ and $A^3{B}^2=A^2{B}^3$, then the value of the determinant of the matrix $A^3+B^3$ is equal to :

If $A=\left[\begin{array}{lll}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$ and $M=A+A^2+A^3+\dots .+A^{20}$, then the sum of all the elements of the matrix $M$ is equal to ______

Consider the system of linear equations
$-x+y+2z=0$
$3x-ay+5z=1$
$2x-2y-az=7$
Let $S_1$ be the set of all $a\in R$ for which the system is inconsistent and $S_2$ be the set of all $\mathrm{a}\in \mathrm{R}$ for which the system has infinitely many solut …

Let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{i}& -\mathrm{i}\\ -\mathrm{i}& \mathrm{i}\end{array}\right],\mathrm{i}=\sqrt{-1}$. Then, the system of linear equation …