Matrices & Determinants · JEE Main previous year questions

Let $a\in R$ and $A$ be a matrix of order $3\times 3$ such that $\det (A)=-4$ and $A+I=\left[\begin{array}{ccc}1& a& 1\\ 2& 1& 0\\ a& 1& 2\end{array}\right]$, where $I$ is the identity matrix of order $3\times 3$. If $\det ((a+1)\mathrm{adj}((a-1)A))$ is $2^m{3}^n,\mathrm{m},\mathrm{n}\in \{0,1,2,\dots ,20\}$, then $\mathrm{m}+\mathrm{n}$ is equal to :

1. A.14
2. B.17
3. C.15
4. D.16

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 :

1. A.76
2. B.12
3. C.4
4. D.20

Let $A=[a_{ij}]$ be a $2\times 2$ matrix such that $a_{ij}\in \{0,1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

1. A.$\frac{5}{8}$
2. B.$\frac{1}{4}$
3. C.$\frac{3}{4}$
4. D.$\frac{3}{8}$

Let $A=[a_{ij}]$ be a matrix of order $3\times 3$, with $a_{ij}=(\sqrt{2}{)}^{i+j}$. If the sum of all the elements in the third row of $A^2$ is $\alpha +\beta \sqrt{2}$, $\alpha ,\beta \in \mathbf{Z}$, then $\alpha +\beta$ is equal to :

1. A.$210$
2. B.$280$
3. C.$224$
4. D.$168$

Let M denote the set of all real matrices of order $3\times 3$ and let $\mathrm{S}=\{-3,-2,-1,1,2\}$. Let ${\mathrm{S}}_1=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:\mathrm{A}={\mathrm{A}}^{\mathrm{T}}\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$, ${\mathrm{S}}_2=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:\mathrm{A}=-{\mathrm{A}}^{\mathrm{T}}\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$, ${\mathrm{S}}_3=\{\mathrm{A}=[a_{ij}]\in \mathrm{M}:a_{11}+a_{22}+a_{33}=0\text{ and }{a}_{ij}\in \mathrm{S},\forall i,j\}$. If $n\left({\mathrm{S}}_1\cup {\mathrm{S}}_2\cup {\mathrm{S}}_3\right)=125\alpha$, then $\alpha$ equals ____ .

Let A be a $3\times 3$ matrix such that ${\mathrm{X}}^{\mathrm{T}}\mathrm{AX}=\mathrm{O}$ for all nonzero $3\times 1$ matrices $\mathrm{X}=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$. If $\mathrm{A}\left[\begin{array}{l}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}1\\ 4\\ -5\end{array}\right],\mathrm{A}\left[\begin{array}{l}1\\ 2\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 4\\ -8\end{array}\right]$, and $\det (\mathrm{adj}(2(\mathrm{A}+\mathrm{I})))=2^{\alpha }{3}^{\beta }{5}^{\gamma },\alpha ,\beta ,\gamma \in N$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is

Let $\mathrm{A}=[{\mathrm{a}}_{\mathrm{ij}}]$ be a square matrix of order $2$ with entries either $0$ or $1$ . Let $\mathrm{E}$ be the event that $\mathrm{A}$ is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is :

1. A.$\frac{3}{8}$
2. B.$\frac{1}{8}$
3. C.$\frac{3}{16}$
4. D.$\frac{5}{8}$

If the system of equations $x+2y-3z=2$ $2x+\lambda y+5z=5$ $14x+3y+\mu z=33$ has infinitely many solutions, then $\lambda +\mu$ is equal to :

1. A.13
2. B.10
3. C.12
4. D.11

Let $A=[a_{ij}]$ be a $3\times 3$ matrix such that $A\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$, $A\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$ and $A\left[\begin{array}{c}2\\ 1\\ 2\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, then $a_{23}$ equals :

1. A.2
2. B.$-1$
3. C.1
4. D.0

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

Let $\lambda ,\mu \in \mathbf{R}$. If the system of equations $3x+5y+\lambda z=3$ $7x+11y-9z=2$ $97x+155y-189z=\mu$ has infinitely many solutions, then $\mu +2\lambda$ is equal to :

1. A.24
2. B.25
3. C.27
4. D.22

Let $\overrightarrow{OA}=2\vec{a}$, $\overrightarrow{OB}=6\vec{a}+5\vec{b}$ and $\overrightarrow{OC}=3\vec{b}$, where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is $15\text{ sq. units}$, then the area (in sq. units) of the quadrilateral $OABC$ is equal to:

1. A.32
2. B.38
3. C.35
4. D.40

Let $A$ be a non-singular matrix of order 3. If $\det (3\mathrm{adj}(2\mathrm{adj}((\det A)A)))=3^{-13}\cdot 2^{-10}$ and $\det (3\mathrm{adj}(2A))=2^m\cdot 3^n$, then $|3m+2n|$ is equal to ______.

Let $B=\left[\begin{array}{cc}1& 3\\ 1& 5\end{array}\right]$ and $A$ be a $2\times 2$ matrix such that $A{B}^{-1}=A^{-1}$. If $BC{B}^{-1}=A$ and $C^4+\alpha C^2+\beta I=O$, then $2\beta -\alpha$ is equal to

1. A.16
2. B.10
3. C.8
4. D.2

Consider the matrices : $A=\left[\begin{array}{cc}2& -5\\ 3& m\end{array}\right],B=\left[\begin{array}{c}20\\ m\end{array}\right]$ and $X=\left[\begin{array}{c}x\\ y\end{array}\right]$. Let the set of all $m$, for which the system of equations $AX=B$ has a negative solution (i.e., $x<0$ and $y<0$), be the interval $(a,b)$. Then $8\underset{a}{\overset{b}{\int }}|A|dm$ is equal to ____.

Let $A=\left[\begin{array}{ccc}2& a& 0\\ 1& 3& 1\\ 0& 5& b\end{array}\right]$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times 3$, then $2a+3b$ is equal to

1. A.$-10$
2. B.$-12$
3. C.$-13$
4. D.$-9$

If $A$ is a square matrix of order 3 such that $\det (A)=3$ and $\det \left(\mathrm{adj}\left(-4\mathrm{adj}\left(-3\mathrm{adj}\left(3\mathrm{adj}\left((2A{)}^{-1}\right)\right)\right)\right)\right)=2^m{3}^n$, then $m+2n$ is equal to :

1. A.2
2. B.4
3. C.3
4. D.6

If the system of equations $2x+7y+\lambda z=3$ $3x+2y+5z=4$ $x+\mu y+32z=-1$ has infinitely many solutions, then $(\lambda -\mu )$ is equal to ____ :

Let $\alpha \beta \ne 0$ and $A=\left[\begin{array}{ccc}\beta & \alpha & 3\\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{array}\right]$. If $B=\left[\begin{array}{ccc}3\alpha & -9& 3\alpha \\ -\alpha & 7& -2\alpha \\ -2\alpha & 5& -2\beta \end{array}\right]$ is the matrix of cofactors of the elements of $A$, then $\det (AB)$ is equal to :

1. A.64
2. B.343
3. C.125
4. D.216

Let $\alpha \in (0,\infty )$ and $A=\left[\begin{array}{ccc}1& 2& \alpha \\ 1& 0& 1\\ 0& 1& 2\end{array}\right]$. If $\det \left(\mathrm{adj}\left(2A-A^T\right)\cdot \mathrm{adj}\left(A-2A^T\right)\right)=2^8$, then $(\det (A){)}^2$ is equal to:

1. A.16
2. B.36
3. C.49
4. D.1

Let $A$ be a $3\times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=3\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$. Then the maximum value of $\det (\mathrm{A})$ is ____.

Let $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ and $B=I+\mathrm{adj}(A)+(\mathrm{adj}A{)}^2+\dots +(\mathrm{adj}A{)}^{10}$. Then, the sum of all the elements of the matrix $B$ is:

1. A.$-110$
2. B.22
3. C.$-124$
4. D.$-88$

Let $A$ be a $2\times 2$ symmetric matrix such that $A\left[\begin{array}{c}1\\ 1\end{array}\right]=\left[\begin{array}{c}3\\ 7\end{array}\right]$ and the determinant of $A$ be 1. If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2\times 2$, then $\alpha +\beta$ equals _________.

Let $A$ be a $3\times 3$ real matrix such that $A\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)=2\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right)=4\left(\begin{array}{c}-1\\ 0\\ 1\end{array}\right),A\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)=2\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$. Then, the system $(A-3I)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)$ has :

1. A.exactly two solutions
2. B.infinitely many solutions
3. C.unique solution
4. D.no solution

Let A be a $3\times 3$ matrix and $\det (A)=2$. If $n=\det (\underset{2024-\text{ times}}{\underbrace{\mathrm{adj}(\mathrm{adj}(\dots ..(\mathrm{adj}A)))}})$, then the remainder when $n$ is divided by 9 is equal to ____.