Matrices — CBSE Class 12 board PYQs

The product of matrix $P$ and $Q$ is equal to a diagonal matrix. If the order of matrix $Q$ is $3\times 2$, then order of matrix $P$ is:

If inverse of matrix $\left[\begin{array}{ccc}7& -3& -3\\ -1& 1& 0\\ -1& 0& 1\end{array}\right]$ is the matrix $\left[\begin{array}{lll}1& 3& 3\\ 1& \lambda & 3\\ 1& 3& 4\end{array}\right]$, then value of $\lambda$ is:

Find the matrix $A^2$, where $A=[a_{ij}]$ is a $2\times 2$ matrix whose elements are given by $a_{ij}=\max (i,j)-\min (i,j)$:

If $A$ is a square matrix of order $3$ such that the value of $|\mathrm{adj}A|=8$, then the value of $|A^{\top }|$ is:

If $A$ is a $2\times 3$ matrix such that $AB$ and $A{B}^{\prime }$ both are defined, then order of matrix $B$ is:

If $\left[\begin{array}{ll}2& 0\\ 5& 4\end{array}\right]=P+Q$ where $P$ is a symmetric and $Q$ is a skew symmetric matrix, then $Q$ is equal to:

If $A$ is a square matrix such that $A^2=A$, then $(I-A{)}^3+A$ is equal to:

If $A$ is a matrix of order $3\times 2$, then the order of the matrix $A^{\prime }$ is ____.
OR
A square matrix $A$ is said to be skew-symmetric, if ____.

If $A=\left[\begin{array}{ccc}2& 0& 0\\ -1& 2& 3\\ 3& 3& 5\end{array}\right]$, then find $A(\mathrm{adj}A)$.

If the matrix $A=\left[\begin{array}{ccc}0& a& -3\\ 2& 0& -1\\ b& 1& 0\end{array}\right]$ is skew symmetric, find the values of $a$ and $b$.

Given $A=\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$, compute $A^{-1}$ and show that $2A^{-1}=9I-A$.

Matrix $A=\left[\begin{array}{ccc}0& 2b& -2\\ 3& 1& 3\\ 3a& 3& -1\end{array}\right]$ is given to be symmetric; find the values of $a$ and $b$.

If $A$ is a square matrix such that $A^2=I$, then find the simplified value of $(A-I{)}^3+(A+I{)}^3-7A$.

Write the element $a_{23}$ of a $3\times 3$ matrix $A=(a_{ij})$ whose elements $a_{ij}$ are given by $a_{ij}=\frac{|i-j|}{2}$.

If $A=\left(\begin{array}{ccc}2& 0& 1\\ 2& 1& 3\\ 1& -1& 0\end{array}\right)$, find $A^2-5A+4I$ and hence find a matrix $X$ such that $A^2-5A+4I+X=0$.

If $A$ is a square matrix such that $A^2=A$, then write the value of $7A-(I+A{)}^3$, where $I$ is the identity matrix.

If $\left[\begin{array}{ll}x-y& z\\ 2x-y& w\end{array}\right]=\left[\begin{array}{cc}-1& 4\\ 0& 5\end{array}\right]$, find the value of $x+y$.

If $\left[\begin{array}{ccc}9& -1& 4\\ -2& 1& 3\end{array}\right]=A+\left[\begin{array}{ccc}1& 2& -1\\ 0& 4& 9\end{array}\right]$, then find the matrix $A$.

Find the value of $a$ if $\left[\begin{array}{cc}a-b& 2a+c\\ 2a-b& 3c+d\end{array}\right]=\left[\begin{array}{cc}-1& 5\\ 0& 13\end{array}\right]$.

If $\left(\begin{array}{cc}2& 3\\ 5& 7\end{array}\right)\left(\begin{array}{cc}1& -3\\ -2& 4\end{array}\right)=\left(\begin{array}{ll}-4& 6\\ -9& x\end{array}\right)$, write the value of $x$.

Simplify: $\cos \theta \left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]+\sin \theta \left[\begin{array}{cc}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{array}\right]$.

For a $2\times 2$ matrix $A=[a_{ij}]$ whose elements are given by $a_{ij}=\frac{i}{j}$, write the value of $a_{12}$.

Write $A^{-1}$ for $A=\left[\begin{array}{ll}2& 5\\ 1& 3\end{array}\right]$.

If $A=\left[\begin{array}{cc}2& 3\\ 5& -2\end{array}\right]$, write $A^{-1}$ in terms of $A$.

If a matrix has $5$ elements, write all possible orders it can have.

If $A=\left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)$, then for what value of $\alpha$ is $A$ an identity matrix?

If $\left(\begin{array}{ll}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{ll}3& 1\\ 2& 5\end{array}\right)=\left(\begin{array}{ll}7& 11\\ k& 23\end{array}\right)$, then write the value of $k$.

If matrix $A=\left[\begin{array}{lll}1& 2& 3\end{array}\right]$, write $A{A}^{\prime }$, where $A^{\prime }$ is the transpose of matrix $A$.

If $A$ is an invertible matrix of order $3$ and $|A|=5$, find $|\mathrm{adj}A|$.

Find the value of $x$ if $\left(\begin{array}{ll}3x+y& -y\\ 2y-x& 3\end{array}\right)=\left(\begin{array}{ll}1& 2\\ -5& 3\end{array}\right)$.

Find the value of $x$ and $y$ if: $2\left[\begin{array}{ll}1& 3\\ 0& x\end{array}\right]+\left[\begin{array}{ll}y& 0\\ 1& 2\end{array}\right]=\left[\begin{array}{ll}5& 6\\ 1& 8\end{array}\right]$.

If $\left[\begin{array}{ll}x+3y& y\\ 7-x& 4\end{array}\right]=\left[\begin{array}{rr}4& -1\\ 0& 4\end{array}\right]$, find the value of $x$ and $y$.

If $A=\left[\begin{array}{l}3\\ 5\\ 2\end{array}\right]$ and $B=\left[\begin{array}{lll}1& 0& 4\end{array}\right]$, find $AB$.

If $A=\left[\begin{array}{lll}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$, verify that $A^2=I$.

Find the values of $a$ and $b$ for which the following holds:

If $A=\left[\begin{array}{c}-2\\ 4\\ 5\end{array}\right]$ and $B=\left[\begin{array}{lll}1& 3& -6\end{array}\right]$, find $B^{\prime }{A}^{\prime }$.

If $A=\left[\begin{array}{ll}2& 3\\ 1& 2\end{array}\right]$, prove that $A^3-4A^2+A=0$.

If $A=\left[\begin{array}{cc}2& -1\\ 3& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 4\\ -1& 7\end{array}\right]$, find $3A^2-2B+I$.

If $A=\left[\begin{array}{cc}1& 0\\ -1& 7\end{array}\right]$, find $K$ such that $A^2-8A+KI=0$.

If $A=\left(\begin{array}{ccc}1& -2& 3\\ 0& -1& 4\\ -2& 2& 1\end{array}\right)$, find $(A^{\prime }{)}^{-1}$.

If $A=\left(\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right)$, find $\alpha$ satisfying $0<\alpha <\frac{\pi }{2}$ when $A+A^{\top }=\sqrt{2}{I}_2$, where $A^{\top }$ is the transpose of $A$.

If $A=\left[\begin{array}{cc}4& 2\\ -1& 1\end{array}\right]$, show that $(A-2I)(A-3I)=0$.

If $A=\left[\begin{array}{ccc}1& 2& -3\\ 2& 3& 2\\ 3& -3& -4\end{array}\right]$, then find $A^{-1}$ and hence solve the following system of equations:

If $A=\left[\begin{array}{ccc}-3& -2& -4\\ 2& 1& 2\\ 2& 1& 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 0\\ -2& -1& -2\\ 0& -1& 1\end{array}\right]$, then find $AB$ and use it to solve the following system of equations: …

If $f(\alpha )=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]$, then prove that $f(\alpha )\cdot f(-\beta )=f(\alpha -\beta )$.

If $A=\left[\begin{array}{ccc}1& 2& -3\\ 3& 2& -2\\ 2& -1& 1\end{array}\right]$, then find $A^{-1}$ and use it to solve the following system of equations:

OR
Using properties of determinants, prove that …

Find the inverse of the following matrix using elementary transformations:

Using the matrix method, solve the following system of equations:

If $A=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$, find $A^{-1}$. Use it to solve the system of equations

Using elementary row transformations, find the inverse of the matrix $A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right]$.

Show that all the diagonal elements of a skew symmetric matrix are zero.

Use the product $\left[\begin{array}{ccc}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right]\left[\begin{array}{ccc}-2& 0& 1\\ 9& 2& -3\\ 6& 1& -2\end{array}\right]$ to solve the system of equations

A typist charges Rs. 145 for typing 10 English and 3 Hindi pages, while the charge for typing 3 English and 10 Hindi pages is Rs. 180. Using matrices, find the charge of typing one English and one Hindi page separately. The typist charged only Rs. 2 per page from a poor student S …

The monthly incomes of Aryan and Babban are in the ratio $3:4$ and their monthly expenditures are in the ratio $5:7$. If each saves Rs. 15,000 per month, find their monthly incomes using the matrix method. This problem reflects which value?

Using elementary transformations, find the inverse of the matrix $A=\left(\begin{array}{lll}8& 4& 3\\ 2& 1& 1\\ 1& 2& 2\end{array}\right)$ and use it to solve the following system of linear equations:

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand-made fans, mats and plates from recycled material at a cost of Rs. 25, Rs. 100 and Rs. 50 each. The number of articles sold are: hand-fans A $40$, B $25$ …

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A awards Rs. $x$, Rs. $y$, Rs. $z$ each for the three values to 3, 2 and 1 students respectively with a total of Rs. 1,600. School B awards 4, 1 and 3 studen …

Using elementary operations, find the inverse of the following matrix:

A school wants to award its students for the values of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for Hard work added to that given for Honesty amounts to Rs. 11,000. The award money given for Honesty and Hard work together …

Using matrices, solve the following system of linear equations:

Using the matrix method, solve the following system of equations:

Using elementary transformations, find the inverse of the matrix $\left(\begin{array}{ccc}1& 3& -2\\ -3& 0& -1\\ 2& 1& 0\end{array}\right)$.

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Using elementary row operations, find the inverse of the matrix $A=\left[\begin{array}{ll}2& 5\\ 1& 3\end{array}\right]$.

Using matrices, solve the following system of equations:

Let $A=\left[\begin{array}{lll}3& 2& 5\\ 4& 1& 3\\ 0& 6& 7\end{array}\right]$. Express $A$ as the sum of two matrices such that one is symmetric and the other is skew symmetric.

If $A=\left[\begin{array}{lll}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$, verify that $A^2-4A-5I=0$.

If $A=\left[\begin{array}{cc}2& -3\\ 3& 4\end{array}\right]$, show that $A^2-6A+17I=0$. Hence find $A^{-1}$.

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Express the following matrix as the sum of a symmetric and a skew symmetric matrix.

If $A=\left[\begin{array}{cc}3& -4\\ -1& 2\end{array}\right]$, find a matrix $B$ such that $AB=I$.

If $A=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right]$, show by induction that $A^n=\left[\begin{array}{cc}\cos n\theta & \sin n\theta \\ -\sin n\theta & \cos n\theta \end{array}\right]$.

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Using matrices, solve the following system of equations:

Using matrices, solve the following system of linear equations:

If $A=\left[\begin{array}{ll}3& 1\\ 7& 5\end{array}\right]$, find $x$ and $y$ such that $A^2+xI=yA$. Hence find $A^{-1}$.

Let $A=\left(\begin{array}{cc}2& -1\\ 3& 4\end{array}\right)$, $B=\left(\begin{array}{ll}5& 2\\ 7& 4\end{array}\right)$, $C=\left(\begin{array}{ll}2& 5\\ 3& 8\end{array}\right)$. Find a matrix $D$ such that $CD-AB=O$.

If $A=\left(\begin{array}{ccc}1& 0& 2\\ 0& 2& 1\\ 2& 0& 3\end{array}\right)$ and $A^3-6A^2+7A+k{I}_3=O$, find $k$.

If $A=\left[\begin{array}{lll}1& 3& 4\\ 2& 1& 2\\ 5& 1& 1\end{array}\right]$, find $A^{-1}$. Hence solve the system of equations

Show that for the matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& -3\\ 2& -1& 3\end{array}\right]$, $A^3-6A^2+5A+11I=0$. Hence find $A^{-1}$.