Matrices & Determinants · JEE Main previous year questions

Let $P=\left[\begin{array}{ccc}-30& 20& 56\\ 90& 140& 112\\ 120& 60& 14\end{array}\right]$ and $A=\left[\begin{array}{ccc}2& 7& {\omega }^2\\ -1& -\omega & 1\\ 0& -\omega & -\omega +1\end{array}\right]$ where $\omega =\frac{-1+i\sqrt{3}}{2}$, and $I_3$ be the identity matrix of order 3. If the determinant of the matrix $(P^{-1}AP-I_3{)}^2$ is $\alpha {\omega }^2$, then the value of $\alpha$ is equal to _________.

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 :

1. A.6
2. B.4
3. C.1
4. D.12

Consider the following system of equations : $x+2y-3z=a$ $2x+6y-11z=b$ $x-2y+7z=c$, where a, b and c are real constants. Then the system of equations :

1. A.has no solution for all a, b and c
2. B.has a unique solution when $5a=2b+c$
3. C.has infinite number of solutions when $5a=2b+c$
4. D.has a unique solution for all a, b and c

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

If for the matrix, $A=\left[\begin{array}{cc}1& -\alpha \\ \alpha & \beta \end{array}\right]$, $A{A}^T=I_2$, then the value of ${\alpha }^4+{\beta }^4$ is :

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

The following system of linear equations $2x+3y+2z=9$ $3x+2y+2z=9$ $x-y+4z=8$

1. A.does not have any solution
2. B.has a solution $(\alpha ,\beta ,\gamma )$ satisfying $\alpha +{\beta }^2+{\gamma }^3=12$
3. C.has a unique solution
4. D.has infinitely many solutions

Let A be a $3\times 3$ matrix with $\det (A)=4$. Let $R_i$ denote the $i^{\text{th}}$ row of A. If a matrix B is obtained by performing the operation $R_2\to 2R_2+5R_3$ on 2A, then $\det (B)$ is equal to :

1. A.64
2. B.16
3. C.128
4. D.80

Let A be a $3\times 3$ matrix such that $\mathrm{adj}\mathrm{A}=\left[\begin{array}{ccc}2& -1& 1\\ -1& 0& 2\\ 1& -2& -1\end{array}\right]$ and $\mathrm{B}=\mathrm{adj}(\mathrm{adj}\mathrm{A})$. If $|\mathrm{A}|=\lambda$ and $|({\mathrm{B}}^{-1}{)}^{\mathrm{T}}|=\mu$, then the ordered pair, $(|\lambda |,\mu )$ is equal to :

1. A.$(3,81)$
2. B.$\left(9,\frac{1}{9}\right)$
3. C.$\left(3,\frac{1}{81}\right)$
4. D.$\left(9,\frac{1}{81}\right)$

Let A be a $2\times 2$ real matrix with entries from $\{0,1\}$ and $|\mathrm{A}|\ne 0$. Consider the following two statements : (P) If $\mathrm{A}\ne {\mathrm{I}}_2$ , then $|\mathrm{A}|=-1$ (Q) If $|\mathrm{A}|=1$, then $\mathrm{tr}(\mathrm{A})=2$, where ${\mathrm{I}}_2$ denotes $2\times 2$ identity matrix and $\mathrm{tr}(\mathrm{A})$ denotes the sum of the diagonal entries of A. Then :

1. A.(P) is true and (Q) is false
2. B.Both (P) and (Q) are false
3. C.Both (P) and (Q) are true
4. D.(P) is false and (Q) is true

Let $a,b,c\in \mathbb{R}$ be all non-zero and satisfy $a^3+b^3+c^3=2$. If the matrix $$A=\left(\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right)$$ satisfies $A^{T}A=I$, then a value of $abc$ can be :

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

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 :

1. A.8
2. B.2
3. C.72
4. D.16

The following system of linear equations 7x + 6y – 2z = 0 3x + 4y + 2z = 0 x – 2y – 6z = 0, has

1. A.no solution
2. B.infinitely many solutions, (x, y, z) satisfying y = 2z
3. C.infinitely many solutions, (x, y, z) satisfying x = 2z
4. D.only the trivial solution

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

1. A.(1, 0)
2. B.(4, 3)
3. C.(4, 6)
4. D.(3, 4)

The number of all $3\times 3$ matrices A, with enteries from the set $\{-1,0,1\}$ such that the sum of the diagonal elements of $\mathrm{A}{\mathrm{A}}^{\mathrm{T}}$ is 3, is

The system of linear equations $$\begin{array}{rl}\lambda x+2y+2z& =5\\ 2\lambda x+3y+5z& =8\\ 4x+\lambda y+6z& =10\end{array}$$ has

1. A.a unique solution when $\lambda =-8$
2. B.no solution when $\lambda =2$
3. C.infinitely many solutions when $\lambda =2$
4. D.no solution when $\lambda =8$

Let $\alpha$ be a root of the equation $x^2+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}1& 1& 1\\ 1& \alpha & {\alpha }^2\\ 1& {\alpha }^2& {\alpha }^4\end{array}\right]$, then the matrix $A^{31}$ is equal to

1. A.$A^2$
2. B.A
3. C.$I_3$
4. D.$A^3$

If the system of linear equations $2x+2ay+az=0$ $2x+3by+bz=0$ $2x+4cy+cz=0$, where a, b, $c\in \mathbb{R}$ are non-zero distinct; has a non-zero solution, then:

1. A.$\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
2. B.$a+b+c=0$
3. C.a, b, c are in G.P.
4. D.a,b,c are in A.P.

Let $A=[a_{ij}]$ and $B=[b_{ij}]$ be two $3\times 3$ real matrices such that $b_{ij}=(3{)}^{(i+j-2)}{a}_{ji}$, where $i,j=1,2,3$. If the determinant of B is 81, then the determinant of A is:

1. A.3
2. B.$\frac{1}{3}$
3. C.$\frac{1}{9}$
4. D.$\frac{1}{81}$

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 :

1. A.$\left[\begin{array}{cc}4& -2\\ 1& -4\end{array}\right]$
2. B.$\left[\begin{array}{cc}-4& -2\\ -1& 4\end{array}\right]$
3. C.$\left[\begin{array}{cc}-4& 2\\ 1& 4\end{array}\right]$
4. D.$\left[\begin{array}{cc}4& -2\\ -1& -4\end{array}\right]$

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 :

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

Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number of beams is :

1. A.180
2. B.210
3. C.170
4. D.190

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

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

Let the number 2,b,c be in an A.P. and $A=\left[\begin{array}{ccc}1& 1& 1\\ 2& b& c\\ 4& b^2& c^2\end{array}\right]$. If $\det (A)\in [2,16]$, then c lies in the interval :

1. A.[2, 3)
2. B.[4, 6]
3. C.$(2+2^{3/4},4)$
4. D.$[3,2+2^{3/4}]$

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

1. A.15
2. B.9
3. C.135
4. D.10

If $A=\left[\begin{array}{ccc}1& \sin \theta & 1\\ -\sin \theta & 1& \sin \theta \\ -1& -\sin \theta & 1\end{array}\right]$; then for all $\theta \in \left(\frac{3\pi }{4},\frac{5\pi }{4}\right)$, $\det (A)$ lies in the interval :

1. A.$(\frac{3}{2},3]$
2. B.$(0,\frac{3}{2}]$
3. C.$[\frac{5}{2},4)$
4. D.$(1,\frac{5}{2}]$