Matrices & Determinants · JEE Main previous year questions

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix A such that $A^2=3A+\alpha I$. If $A^4=21A+\beta I$, then

1. A.$\alpha =1$
2. B.$\alpha =4$
3. C.$\beta =8$
4. D.$\beta =-8$

The set of all values of $\mathrm{t}\in \mathbb{R}$, for which the matrix $$\left[\begin{array}{ccc}{e}^t& e^{-t}(\sin t-2\cos t)& e^{-t}(-2\sin t-\cos t)\\ e^t& e^{-t}(2\sin t+\cos t)& e^{-t}(\sin t-2\cos t)\\ e^t& e^{-t}\cos t& e^{-t}\sin t\end{array}\right]$$ is invertible, is :

1. A.$\{k\pi ,k\in \mathbb{Z}\}$
2. B.$\mathbb{R}$
3. C.$\left\{(2k+1)\frac{\pi }{2},k\in \mathbb{Z}\right\}$
4. D.$\left\{k\pi +\frac{\pi }{4},k\in \mathbb{Z}\right\}$

Let A be a symmetric matrix such that $|\mathrm{A}|=2$ and $\left[\begin{array}{cc}2& 1\\ 3& \frac{3}{2}\end{array}\right]A=\left[\begin{array}{cc}1& 2\\ \alpha & \beta \end{array}\right]$. If the sum of the diagonal elements of A is $s$, then $\frac{\beta s}{{\alpha }^2}$ is equal to ____.

Let $x,y,z>1$ and $A=\left[\begin{array}{ccc}1& {\log }_{x}y& {\log }_{x}z\\ {\log }_{y}x& 2& {\log }_{y}z\\ {\log }_{z}x& {\log }_{z}y& 3\end{array}\right]$. Then $|\mathrm{adj}(\mathrm{adj}{A}^2)|$ is equal to

1. A.$6^4$
2. B.$2^8$
3. C.$4^8$
4. D.$2^4$

Let A, B, C be $3\times 3$ matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements (S1) ${\mathrm{A}}^{13}{\mathrm{B}}^{26}-{\mathrm{B}}^{26}{\mathrm{A}}^{13}$ is symmetric (S2) ${\mathrm{A}}^{26}{\mathrm{C}}^{13}-{\mathrm{C}}^{13}{\mathrm{A}}^{26}$ is symmetric Then,

1. A.Only S2 is true
2. B.Only S1 is true
3. C.Both S1 and S2 are false
4. D.Both S1 and S2 are true

Let $A=\left[\begin{array}{cc}\frac{1}{\sqrt{10}}& \frac{3}{\sqrt{10}}\\ \frac{-3}{\sqrt{10}}& \frac{1}{\sqrt{10}}\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -i\\ 0& 1\end{array}\right]$, where $i=\sqrt{-1}$. If $\mathrm{M}={\mathrm{A}}^{\mathrm{T}}\mathrm{BA}$, then the inverse of the matrix $\mathrm{A}{\mathrm{M}}^{2023}{\mathrm{A}}^{\mathrm{T}}$ is

1. A.$\left[\begin{array}{cc}1& -2023i\\ 0& 1\end{array}\right]$
2. B.$\left[\begin{array}{cc}1& 0\\ 2023i& 1\end{array}\right]$
3. C.$\left[\begin{array}{cc}1& 2023i\\ 0& 1\end{array}\right]$
4. D.$\left[\begin{array}{cc}1& 0\\ -2023i& 1\end{array}\right]$

If A and B are two non-zero $n\times n$ matrices such that $A^2+B=A^{2}B$, then :

1. A.$A^{2}B=I$
2. B.$A^2=I$ or $B=I$
3. C.$A^{2}B=B{A}^2$
4. D.$AB=I$

Let $\alpha$ be a root of the equation $(a-c)x^2+(b-a)x+(c-b)=0$ where a, b, c are distinct real numbers such that the matrix $\left[\begin{array}{ccc}{\alpha }^2& \alpha & 1\\ 1& 1& 1\\ a& b& c\end{array}\right]$ is singular. Then, the value of $\frac{(a-c{)}^2}{(b-a)(c-b)}+\frac{(b-a{)}^2}{(a-c)(c-b)}+\frac{(c-b{)}^2}{(a-c)(b-a)}$ is

1. A.3
2. B.6
3. C.12
4. D.9

If the system of equations $x+2y+3z=3$ $4x+3y-4z=4$ $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to :

1. A.$\left(\frac{72}{5},\frac{21}{5}\right)$
2. B.$\left(-\frac{72}{5},-\frac{21}{5}\right)$
3. C.$\left(-\frac{72}{5},\frac{21}{5}\right)$
4. D.$\left(\frac{72}{5},-\frac{21}{5}\right)$

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :

1. A.125
2. B.150
3. C.225
4. D.120

Let A be a $3\times 3$ matrix such that $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}\mathrm{A}))|=12^4$. Then $|{\mathrm{A}}^{-1}\mathrm{adj}\mathrm{A}|$ is equal to

1. A.12
2. B.$2\sqrt{3}$
3. C.1
4. D.$\sqrt{6}$

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

1. A.the system of linear equations $AX=0$ has a unique solution
2. B.the system of linear equations $AX=0$ has infinitely many solutions
3. C.B is an invertible matrix
4. D.$\mathrm{adj}(\mathrm{A})$ is an invertible matrix

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 ____.

Which of the following matrices can NOT be obtained from the matrix $\left[\begin{array}{cc}-1& 2\\ 1& -1\end{array}\right]$ by a single elementary row operation ?

1. A.$\left[\begin{array}{cc}0& 1\\ 1& -1\end{array}\right]$
2. B.$\left[\begin{array}{cc}1& -1\\ -1& 2\end{array}\right]$
3. C.$\left[\begin{array}{cc}-1& 2\\ -2& 7\end{array}\right]$
4. D.$\left[\begin{array}{cc}-1& 2\\ -1& 3\end{array}\right]$

Let the matrix $A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right]$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\mathrm{Adj}(B_{n-1})$ for all $n\ge 1$, then $\det (B_4)$ is equal to :

1. A.$3^{28}$
2. B.$3^{30}$
3. C.$3^{32}$
4. D.$3^{36}$

Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

1. A.$A^4-B^4$ is a symmetric matrix
2. B.$AB-BA$ is a symmetric matrix
3. C.$B^5-A^5$ is a skew-symmetric matrix
4. D.$AB+BA$ is a skew-symmetric matrix

Let $S$ be the set containing all $3\times 3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is 6 is _________.

Let $A=\left(\begin{array}{cc}4& -2\\ \alpha & \beta \end{array}\right)$. If ${\mathrm{A}}^2+\gamma \mathrm{A}+18\mathrm{I}=\mathrm{O}$, then $\det (\mathrm{A})$ is equal to ____.

1. A.$-18$
2. B.$18$
3. C.$-50$
4. D.$50$

Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If $\frac{\det (\mathrm{adj}(\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))))}{(\alpha -\beta {)}^{16}(\beta -\gamma {)}^{16}(\gamma -\alpha {)}^{16}}=2^{32}\times 3^{16}$, then the number of such 3 - tuples $(\alpha ,\beta ,\gamma )$ is ____.

Let A be a $2\times 2$ matrix with $\det (\mathrm{A})=-1$ and $\det ((\mathrm{A}+\mathrm{I})(\mathrm{Adj}(\mathrm{A})+\mathrm{I}))=4$. Then the sum of the diagonal elements of A can be :

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

Let $A=\left[\begin{array}{ccc}1& -2& \alpha \\ \alpha & 2& -1\end{array}\right]$ and $B=\left[\begin{array}{cc}2& \alpha \\ -1& 2\\ 4& -5\end{array}\right]$, $\alpha \in C$. Then the absolute value of the sum of all values of $\alpha$ for which $\det (AB)=0$ is :

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

Let A and B be two square matrices of order 2. If $\det (A)=2$, $\det (B)=3$ and $\det \left(\left(\det (5(\det A)B)\right)A^2\right)=2^a{3}^b{5}^c$ for some a, b, c, $\in \mathrm{N}$, then $a+b+c$ is equal to :

1. A.10
2. B.12
3. C.13
4. D.14

If the system of linear equations $2x-3y=\gamma +5$, $\alpha x+5y=\beta +1$, where $\alpha ,\beta ,\gamma \in \mathbb{R}$ has infinitely many solutions then the value of $|9\alpha +3\beta +5\gamma |$ is equal to ________.

Let the system of linear equations $$\begin{array}{l}x+2y+z=2,\\ \alpha x+3y-z=\alpha ,\\ -\alpha x+y+2z=-\alpha \end{array}$$ be inconsistent. Then $\alpha$ is equal to :

1. A.$\frac{5}{2}$
2. B.$-\frac{5}{2}$
3. C.$\frac{7}{2}$
4. D.$-\frac{7}{2}$

$x={\sum }_{n=0}^{\infty }{a}^n,y={\sum }_{n=0}^{\infty }{b}^n,z={\sum }_{n=0}^{\infty }{c}^n$, where $a,b,c$ are in A.P. and $|a|<1,|b|<1,|c|<1,abc\ne 0$, then :

1. A.$x,y,z$ are in A.P.
2. B.$x,y,z$ are in G.P.
3. C.$\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.
4. D.$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1-(a+b+c)$