Matrices and Determinants

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)...

Let \(A\) be a \(2 \times 2\) real matrix and \(I\) be the identity matrix of order 2. If the roots of the equation \(|\mathrm{A}-x \mathrm{I}|=0\) be \(-1\) and 3, then the sum of the diagonal elements of the matrix \(\mathrm{A}^2\) is

The values of \(\alpha\), for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval

If the system of equations
$$ \begin{aligned} & 2 x+3 y-z=5 \\\\ & x+\alpha y+3 z=-4 \\\\ & 3 x-y+\beta z=7 \end{aligned} $$
has infinitely many solutions, then $13 \alpha \beta$ is equal to :

If $\mathrm{A}=\left[\begin{array}{cc}\sqrt{2} & 1 \\ -1 & \sqrt{2}\end{array}\right], \mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right], \mathrm{C}=\mathrm{ABA}^{\mathrm{T}}$ and $\mathrm{X}=\mathrm{A}^{\mathrm{T}} \mathrm...

Let $A=I_2-2 M M^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M=I_1$ holds. If $\lambda$ is a real number such that the relation $A X=\lambda X$ holds for some non-zero real matrix $X$ of order $2 \times ...

Let the system of equations $x+2 y+3 z=5,2 x+3 y+z=9,4 x+3 y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2 \mu$ is equal to :

Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$$ and \(Q=P A P^{T}\). If $$P^{T} Q^{2007} P=\left[\begin{arra...

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

Let S be the set of all values of \(\theta \in[-\pi, \pi]\) for which the system of linear equations
\(x+y+\sqrt{3} z=0\)
\(-x+(\tan \theta) y+\sqrt{7} z=0\)
\(x+y+(\tan \theta) z=0\)
has non-trivial solution. Then $$\frac{120}{\pi} \sum_\l...

If $$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$$ and \(\alpha+\beta=-2\), then \(4 \alpha^{2}+\beta^{2}+\lambda^{2}\) is equal to :

Let \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}\), where \(\mathrm{a}_{\mathrm{ij}} \neq 0\) for all \(\mathrm{i}, \mathrm{j}\) and \(\mathrm{A}^{2}=\mathrm{I}\). Let a be the sum of all diagonal elements of $$\mathrm{A}...

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

For the system of equations
\(x+y+z=6\)
\(x+2 y+\alpha z=10\)
\(x+3 y+5 z=\beta\), which one of the following is NOT true?

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

Let \(A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 1} \cr 0 & {12} & { - 3} \cr } } \right)\). Then the sum of the diagonal elements of the matrix \({(A + I)^{11}}\) is equal to :

For the system of linear equations
\(x+y+z=6\)
\(\alpha x+\beta y+7 z=3\)
\(x+2 y+3 z=14\)
which of the following is NOT true ?

Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix

$\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right) \cdot$ is $2^{84}$, then $\mathrm{n}$ is equal to :

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

Let the system of linear equations
\(x+y+kz=2\)
\(2x+3y-z=1\)
\(3x+4y+2z=k\)
have infinitely many solutions. Then the system
\((k+1)x+(2k-1)y=7\)
\((2k+1)x+(k+5)y=10\)
has :

If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then :

For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations
$$ \begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned} $$
has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of ...

Consider the following system of equations
\(\alpha x+2y+z=1\)
\(2\alpha x+3y+z=1\)
\(3x+\alpha y+2z=\beta\)
for some \(\alpha,\beta\in \mathbb{R}\). Then which of the following is NOT correct.

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

Let A be a symmetric matrix such that \(\mathrm{|A|=2}\) and \(\left[ {\matrix{ 2 & 1 \cr 3 & {{3 \over 2}} \cr } } \right]A = \left[ {\matrix{ 1 & 2 \cr \alpha & \beta \cr } } \right]\). If the sum of the diagonal...

The set of all values of \(\mathrm{t\in \mathbb{R}}\), for which the matrix $$\left[ {\matrix{
{{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t}}( - 2\sin t - \cos t)} \cr
{{e^t}} & {{e^{ - t}}(2\sin t + \cos t)} & {{e^{ - t}}(\...

Let \(\mathrm{A_1,A_2,A_3}\) be the three A.P. with the same common difference d and having their first terms as \(\mathrm{A,A+1,A+2}\), respectively. Let a, b, c be the \(\mathrm{7^{th},9^{th},17^{th}}\) terms of \(\mathrm{A_1,A_2,A_3}\), ...

Let S\(_1\) and S\(_2\) be respectively the sets of all \(a \in \mathbb{R} - \{ 0\}\) for which the system of linear equations
\(ax + 2ay - 3az = 1\)
\((2a + 1)x + (2a + 3)y + (a + 1)z = 2\)
\((3a + 5)x + (a + 5)y + (a + 2)z = 3\)
has uniq...

Let \(x,y,z > 1\) and \(A = \left[ {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 2 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 3 \cr } } \right]\). Then \(\mathrm{|adj~(adj~A^2)|}\) is equal ...

Let \(A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]\) and $$B = \left[ {\matrix{
1 & { - i} \cr
0 & 1 \cr

} ...

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) A\(^{13}\) B\(^{26}\) \(-\) B\(^{26}\) A\(^{13}\) is symmetric
(S2) A\(^{26}\) C\(^{13}\) \(-\) C\(^{13}\) A$$^{26}...

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[ {\matrix{
{{\alpha ^2}} & \alpha & 1 \cr
1 & 1 & 1 \cr
a & b & c \cr

...

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

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 :

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

Let \(S\) denote the set of all real values of \(\lambda\) such that the system of equations
\(\lambda x+y+z=1\)
\(x+\lambda y+z=1\)
\(x+y+\lambda z=1\)
is inconsistent, then $$\sum_\limits{\lambda \in S}\left(|\lambda|^{2}+|\lambda|\right)...

If \(A = {1 \over 2}\left[ {\matrix{ 1 & {\sqrt 3 } \cr { - \sqrt 3 } & 1 \cr } } \right]\), then :

For the system of linear equations \(\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta\), which one of the following statements is NOT correct?

Let the determinant of a square matrix A of order $m$ be $m-n$, where $m$ and $n$ satisfy $4 m+n=22$ and $17 m+4 n=93$. If $\operatorname{det}(n \operatorname{adj}(\operatorname{adj}(m A)))=3^{a} 5^{b} 6^{c}$ then $a+b+c$ is equal to :

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 :

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

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

If the system of equations
\(2 x+y-z=5\)
\(2 x-5 y+\lambda z=\mu\)
\(x+2 y-5 z=7\)
has infinitely many solutions, then \((\lambda+\mu)^{2}+(\lambda-\mu)^{2}\) is equal to

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

Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^{2}+n+2 & n^{2} \\ n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If \(\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96\), then \(n\) is equal to _____________.

Let $$A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$$, then the sum of all the el...

Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where \(a, c \in \mathbb{R}\). If \(A^{3}=A\) and the positive value of \(a\) belongs to the interval \((n-1, n]\), where \(n \in \mathbb{N}\), then $$...

Let \(\mathrm{A}\) be a \(2 \times 2\) matrix with real entries such that \(\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If \(\operatorname{det}\left(A^{2}-A\right)=4\), then the sum of all possible ...

$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then \(\lambda, \frac{\lambda}{3}\) are the roots of the equation :

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