Matrices & Determinants · JEE Main previous year questions

Let A be a square matrix such that ${\text{AA}}^{\text{T}}=\text{I}$. Then $\frac{1}{2}\text{A}\left[(\text{A}+{\text{A}}^{\text{T}}{)}^2+(\text{A}-{\text{A}}^{\text{T}}{)}^2\right]$ is equal to

1. A.${\text{A}}^2+{\text{A}}^{\text{T}}$
2. B.${\text{A}}^3+\text{I}$
3. C.${\text{A}}^3+{\text{A}}^{\text{T}}$
4. D.${\text{A}}^2+\text{I}$

Let $\overrightarrow{OA}=\vec{a},\overrightarrow{OB}=12\vec{a}+4\vec{b}$ and $\overrightarrow{OC}=\vec{b}$, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $\frac{\text{area of the quadrilateral OA BC}}{\text{area of S}}$ is equal to _________.

1. A.7
2. B.6
3. C.8
4. D.10

Consider the matrix $f(x)=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$. Given below are two statements : Statement I : $f(-x)$ is the inverse of the matrix $f(x)$. Statement II : $f(x)f(y)=f(x+y)$. In the light of the above statements, choose the correct answer from the options given below :

1. A.Statement I is false but Statement II is true
2. B.Both Statement I and Statement II are false
3. C.Both Statement I and Statement II are true
4. D.Statement I is true but Statement II is false

Let $A=\left[\begin{array}{ccc}2& 0& 1\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]$, $B=[B_1,B_2,B_3]$, where $B_1,B_2,B_3$ are column matrices, and $${\text{AB}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],{\text{AB}}_2=\left[\begin{array}{c}2\\ 3\\ 0\end{array}\right],{\text{AB}}_3=\left[\begin{array}{c}3\\ 2\\ 1\end{array}\right]$$ If $\alpha =|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then ${\alpha }^3+{\beta }^3$ is equal to ________.

Let $A$ be a $2\times 2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $|A-xI|=0$ be $-1$ and $3$, then the sum of the diagonal elements of the matrix $A^2$ is _______.

For the system of linear equations $$2x+4y+2az=b$$ $$x+2y+3z=4$$ $$2x-5y+2z=8$$ which of the following is NOT correct?

1. A.It has infinitely many solutions if $a=3,b=8$
2. B.It has infinitely many solutions if $a=3,b=6$
3. C.It has unique solution if $a=b=8$
4. D.It has unique solution if $a=b=6$

Let $B=\left[\begin{array}{ccc}1& 3& \alpha \\ 1& 2& 3\\ \alpha & \alpha & 4\end{array}\right]$, $\alpha >2$ be the adjoint of a matrix $A$ and $|A|=2$. Then $[\begin{array}{ccc}\alpha & -2\alpha & \alpha \end{array}]B\left[\begin{array}{c}\alpha \\ -2\alpha \\ \alpha \end{array}\right]$ is equal to :

1. A.$32$
2. B.$-16$
3. C.$0$
4. D.$16$

The number of symmetric matrices of order 3, with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$ is :

1. A.$10^9$
2. B.$9^{10}$
3. C.$10^6$
4. D.$6^{10}$

Let for $A=\left[\begin{array}{ccc}1& 2& 3\\ \alpha & 3& 1\\ 1& 1& 2\end{array}\right]$, $|A|=2$. If $|2\mathrm{adj}(2\mathrm{adj}(2\mathrm{A}))|=32^n$, then $3n+\alpha$ is equal to

1. A.$11$
2. B.$9$
3. C.$12$
4. D.$10$

Let $A=\left[\begin{array}{cc}1& \frac{1}{51}\\ 0& 1\end{array}\right]$. If $B=\left[\begin{array}{cc}1& 2\\ -1& -1\end{array}\right]A\left[\begin{array}{cc}-1& -2\\ 1& 1\end{array}\right]$, then the sum of all the elements of the matrix ${\sum }_{n=1}^{50}{B}^n$ is equal to

1. A.$50$
2. B.$75$
3. C.$100$
4. D.$125$

Let $S=\{M=[a_{ij}],a_{ij}\in \{0,1,2\},1\le i,j\le 2\}$ be a sample space and $A=\{M\in S:M\text{ is invertible}\}$ be an event. Then $P(A)$ is equal to :

1. A.$\frac{47}{81}$
2. B.$\frac{49}{81}$
3. C.$\frac{50}{81}$
4. D.$\frac{16}{27}$

Let A be a $2\times 2$ matrix with real entries such that $A^{\prime }=\alpha A+\text{I}$, where $\alpha \in \mathbb{R}-\{-1,1\}$. If $\det (A^2-A)=4$, then the sum of all possible values of $\alpha$ is equal to :

1. A.2
2. B.$\frac{3}{2}$
3. C.0
4. D.$\frac{5}{2}$

If the system of linear equations $$7x+11y+\alpha z=13$$ $$5x+4y+7z=\beta$$ $$175x+194y+57z=361$$ has infinitely many solutions, then $\alpha +\beta +2$ is equal to :

1. A.$6$
2. B.$4$
3. C.$5$
4. D.$3$

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

1. A.$3^{12}\cdot 6^{10}$
2. B.$3^{11}\cdot 6^{10}$
3. C.$3^{12}\cdot 6^{11}$
4. D.$3^{10}\cdot 6^{11}$

If $A=\frac{1}{5!6!7!}\left[\begin{array}{ccc}5!& 6!& 7!\\ 6!& 7!& 8!\\ 7!& 8!& 9!\end{array}\right]$, then $|\mathrm{adj}(\mathrm{adj}(2A))|$ is equal to :

1. A.$2^{12}$
2. B.$2^{20}$
3. C.$2^8$
4. D.$2^{16}$

Let the equations of two adjacent sides of a parallelogram ABCD be $2x-3y=-23$ and $5x+4y=23$. If the equation of its one diagonal AC is $3x+7y=23$ and the distance of A from the other diagonal is d, then $50d^2$ is equal to _________.

Let $A=\left[\begin{array}{ccc}2& 1& 0\\ 1& 2& -1\\ 0& -1& 2\end{array}\right]$. If $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}2A))|=(16{)}^n$, then $n$ is equal to :

1. A.$9$
2. B.$8$
3. C.$10$
4. D.$12$

If the system of equations $$x+y+az=b$$ $$2x+5y+2z=6$$ $$x+2y+3z=3$$ has infinitely many solutions, then $2a+3b$ is equal to :

1. A.28
2. B.25
3. C.20
4. D.23

Let $\text{A}=[{\text{a}}_{\text{ij}}{]}_{2\times 2}$, where ${\text{a}}_{\text{ij}}\ne 0$ for all i, j and ${\text{A}}^2=\text{I}$. Let a be the sum of all diagonal elements of A and $\text{b}=|\text{A}|$. Then $3a^2+4b^2$ is equal to :

1. A.4
2. B.3
3. C.14
4. D.7

Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha ,\beta ,\gamma ,\delta \in \mathbb{N}$, if $P^{\alpha }+P^{\beta }=\gamma I-29P$ and $P^{\alpha }-P^{\beta }=\delta I-13P$, then $\alpha +\beta +\gamma -\delta$ is equal to :

1. A.18
2. B.22
3. C.24
4. D.40

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 ?

1. A.If $\alpha =\beta =7$, then the system has no solution
2. B.For every point $(\alpha ,\beta )\ne (7,7)$ on the line $x-2y+7=0$, the system has infinitely many solutions
3. C.There is a unique point $(\alpha ,\beta )$ on the line $x+2y+18=0$ for which the system has infinitely many solutions
4. D.If $\alpha =\beta$ and $\alpha \ne 7$, then the system has a unique solution

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 :

1. A.4094
2. B.2050
3. C.6144
4. D.4097

Let $\mathrm{A}=[a_{ij}],a_{ij}\in \mathbb{Z}\cap [0,4],1\le i,j\le 2$. The number of matrices A such that the sum of all entries is a prime number $p\in (2,13)$ is ___________.

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

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

1. A.$1+d^2=m^2+q^2$
2. B.$1+d^2=(m+q{)}^2$
3. C.$(1+d{)}^2=m^2+q^2$
4. D.$(1+d{)}^2=(m+q{)}^2$