Matrices & Determinants · JEE Main previous year questions

Let A and B be two invertible matrices of order $3\times 3$. If $\det (\mathrm{AB}{\mathrm{A}}^{\mathrm{T}})=8$ and $\det (\mathrm{A}{\mathrm{B}}^{-1})=8$, then $\det (\mathrm{B}{\mathrm{A}}^{-1}{\mathrm{B}}^{\mathrm{T}})$ is equal to :

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

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then the matrix $A^{-50}$ when $\theta =\frac{\pi }{12}$, is equal to :

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

The number of values of k for which the system of linear equations, (k + 2)x + 10y = k kx + (k +3)y = k -1 has no solution, is :

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

Let A be a matrix such that $A\left[\begin{array}{cc}1& 2\\ 0& 3\end{array}\right]$ is a scalar matrix and $|3A|=108$. Then $A^2$ equals :

1. A.$\left[\begin{array}{cc}4& -32\\ 0& 36\end{array}\right]$
2. B.$\left[\begin{array}{cc}36& 0\\ -32& 4\end{array}\right]$
3. C.$\left[\begin{array}{cc}4& 0\\ -32& 36\end{array}\right]$
4. D.$\left[\begin{array}{cc}36& -32\\ 0& 4\end{array}\right]$

If the system of linear equations $x+ky+3z=0$ $3x+ky-2z=0$ $2x+4y-3z=0$ has a non-zero solution (x, y, z), then $\frac{xz}{y^2}$ is equal to

1. A.$30$
2. B.$-10$
3. C.$10$
4. D.$-30$

For two $3\times 3$ matrices A and B, let $\mathrm{A}+\mathrm{B}=2{\mathrm{B}}^{\mathrm{T}}$ and $3\mathrm{A}+2\mathrm{B}={\mathrm{I}}_3$, where ${\mathrm{B}}^{\mathrm{T}}$ is the transpose of B and ${\mathrm{I}}_3$ is $3\times 3$ identity matrix. Then :

1. A.$5\mathrm{A}+10\mathrm{B}=2{\mathrm{I}}_3$
2. B.$10\mathrm{A}+5\mathrm{B}=3{\mathrm{I}}_3$
3. C.$\mathrm{B}+2\mathrm{A}={\mathrm{I}}_3$
4. D.$3\mathrm{A}+6\mathrm{B}=2{\mathrm{I}}_3$

The two adjacent sides of a cyclic quadrilateral are 2 and 5 and the angle between them is $60^{\circ }$. If the area of the quadrilateral is $4\sqrt{3}$, then the perimeter of the quadrilateral is :

1. A.$12.5$
2. B.$13.2$
3. C.$12$
4. D.$13$

Let $A$ be any $3\times 3$ invertible matrix. Then which one of the following is not always true ?

1. A.$\mathrm{adj}(A)=|A|\cdot A^{-1}$
2. B.$\mathrm{adj}(\mathrm{adj}(A))=|A|\cdot A$
3. C.$\mathrm{adj}(\mathrm{adj}(A))=|A{|}^2\cdot (\mathrm{adj}(A){)}^{-1}$
4. D.$\mathrm{adj}(\mathrm{adj}(A))=|A|\cdot (\mathrm{adj}(A){)}^{-1}$

If $A=\left[\begin{array}{cc}5a& -b\\ 3& 2\end{array}\right]$ and $A\mathrm{adj}A=A{A}^T$, then $5a+b$ is equal to :

1. A.$4$
2. B.$13$
3. C.$-1$
4. D.$5$

If $A=\left[\begin{array}{ccc}1& 2& 2\\ 2& 1& -2\\ a& 2& b\end{array}\right]$ is a matrix satisfying the equation $A{A}^T=9I$, where $I$ is $3\times 3$ identity matrix, then the ordered pair $(a,b)$ is equal to :

1. A.$(2,1)$
2. B.$(-2,-1)$
3. C.$(2,-1)$
4. D.$(-2,1)$

If $A$ is a $3\times 3$ non-singular matrix such that $A{A}^{\prime }=A^{\prime }A$ and $B=A^{-1}{A}^{\prime }$, then $B{B}^{\prime }$ equals:

1. A.$B^{-1}$
2. B.$(B^{-1}{)}^{\prime }$
3. C.$I+B$
4. D.$I$

Statement 1: If the system of equations $x+ky+3z=0,3x+ky-2z=0,2x+3y-4z=0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$. Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

1. A.Statement 1 is false, Statement 2 is true.
2. B.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
3. C.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
4. D.Statement 1 is true, Statement 2 is false.

Let $A$ and $B$ be real matrices of the form $\left[\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right]$ and $\left[\begin{array}{cc}0& \gamma \\ \delta & 0\end{array}\right]$, respectively. Statement 1: $AB-BA$ is always an invertible matrix. Statement 2: $AB-BA$ is never an identity matrix.

1. A.Statement 1 is true, Statement 2 is false.
2. B.Statement 1 is false, Statement 2 is true.
3. C.Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation of Statement 1.
4. D.Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation of Statement 1.

If $A=\left(\begin{array}{c}\alpha -1\\ 0\\ 0\end{array}\right),B=\left(\begin{array}{c}\alpha +1\\ 0\\ 0\end{array}\right)$ be two matrices, then $A{B}^T$ is a non-zero matrix for $|\alpha |$ not equal to

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

Let $A$ be a $2\times 2$ matrix with non-zero entries and let $A^2=I$, where $I$ is $2\times 2$ identity matrix. Define $\mathrm{Tr}(A)=$ sum of diagonal elements of $A$ and $|A|=$ determinant of matrix $A$. Statement- 1: $\mathrm{Tr}(A)=0$. Statement- 2: $|A|=1$ .

1. A.statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
2. B.statement - 1 is true, statement - 2 is false.
3. C.statement - 1 is false, statement -2 is true
4. D.statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

The number of $3\times 3$ non-singular matrices, with four entries as 1 and all other entries as 0, is :

1. A.$5$
2. B.$6$
3. C.at least 7
4. D.less than 4

Let $A$ be a $2\times 2$ matrix Statement - 1 : $adj(adjA)=A$ Statement - 2 : $|adjA|=|A|$

1. A.statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
2. B.statement - 1 is true, statement - 2 is false.
3. C.statement - 1 is false, statement - 2 is true
4. D.statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

If $A$ and $B$ are square matrices of size $n\times n$ such that $A^2-B^2=(A-B)(A+B)$, then which of the following will be always true?

1. A.$A=B$
2. B.$AB=BA$
3. C.either of $A$ or $B$ is a zero matrix
4. D.either of $A$ or $B$ is identity matrix

If $a_1,a_2,a_3,\dots ,a_n,\dots$ are in G.P., then the determinant $\Delta =\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$ is equal to :

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

Let $A=\left(\begin{array}{ccc}0& 0& -1\\ 0& -1& 0\\ -1& 0& 0\end{array}\right)$. The only correct statement about the matrix $A$ is

1. A.$A^2=I$
2. B.$A=(-1)I$, where $I$ is a unit matrix
3. C.$A^{-1}$ does not exist
4. D.$A$ is a zero matrix

Let $A=\left(\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\\ 1& 1& 1\end{array}\right)$ and $10B=\left(\begin{array}{ccc}4& 2& 2\\ -5& 0& \alpha \\ 1& -2& 3\end{array}\right)$. If $B$ is the inverse of matrix $A$, then $\alpha$ is

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

If $a_1,a_2,a_3,\dots \dots \dots ,a_n,\dots \dots$ are in G.P., then the value of the determinant $\left|\begin{array}{ccc}\log a_n& \log a_{n+1}& \log a_{n+2}\\ \log a_{n+3}& \log a_{n+4}& \log a_{n+5}\\ \log a_{n+6}& \log a_{n+7}& \log a_{n+8}\end{array}\right|$, is

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