Matrices practice · Mixed difficulty

Let $A$ and $B$ are the non-singular square matrices, then which of the following is always correct?

Let $\mathrm{A}=\left\{\mathrm{X}=(x, y, z)^{\mathrm{T}}: \mathrm{PX}=0\right.$ and $\left.x^2+y^2+z^2=1\right\}$, where $P=\left[\begin{array}{ccc}1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1\end{array}\right]$, then the set A :

Consider the matrix $A=\left[\begin{array}{cccc}2 & 5 & 19 & -7 \\ 35 & -2 & 5 / 2 & 12 \\ \sqrt{3} & 1 & -5 & 17\end{array}\right]$

Now, consider the following statements

I. The order of the matrix is $4 \times 3$ and number of elements is 12 .

II. The elements ${a}_{13}, {a}_{21}, {a}_{33}$ are respectively $19,35,-5$. Choose the correct option.

The value of $\theta$ lying between $\theta = 0$ and $\frac{\pi }{2}$ and satisfying the equation $\left| {\begin{array}{cccccccccccccccccccc} {1 + {{\sin }^2}\mathrm{\theta }}&{{{\cos }^2}\mathrm{\theta }}&{4\sin 4\mathrm{\theta }}\\ {{{\sin }^2}\mathrm{\theta }}&{1 + {{\cos }^2}\mathrm{\theta }}&{4\sin 4\mathrm{\theta }}\\ {{{\sin }^2}\mathrm{\theta }}&{{{\cos }^2}\mathrm{\theta }}&{1 + 4\sin 4\mathrm{\theta }} \end{array}} \right|$ is

The system of equations $\alpha x + y + z = \alpha - 1,\,\,\,x + \alpha y + z = \alpha - 1,\,\,{\text{ and }}x + y + \alpha z = \alpha - 1{\text{ has no solution, if }}\alpha {\text{ is }}$

Let $A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$. If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj2A}))|=(16)^{n}$, then $n$ is equal to

If $A=\left[\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right]$ and $A B=O$, then $B=$

If the system of equations $2x + 3y - z = 0,x + ky - 2z = 0{\kern 1pt} \,\mathrm{and}\,2x - y + z = 0$ has a non-trivial solution (x, y, z), then $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} + k$ is equal to

The value of $k \in R$, for which the following system of linear equations $3 x-y+4 z=3, x+2 y-3 z=-2,6 x+5 y+k z=-3$ has infinitely many solutions, is

Matrix $A=\left[\begin{array}{ccc}x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z\end{array}\right]$, if $x y z=60$ and $8 x+4 y+3 z$ $=20$, then $A(\operatorname{adj} A)$ is equal to

If $A=\left[\begin{array}{cc}1 & -2 \\ 5 & 3\end{array}\right]$, then $A+A^T$ equals:

For the equations $x + 2y + 3z = 1,\;2x + y + 3z = 2\;$ and $\;5x + 5y + 9z = 4$

If $3 \mathrm{X}+2 \mathrm{Y}=\mathrm{I}$ and $2 \mathrm{X}-\mathrm{Y}=\mathrm{O}$, where $\mathrm{I}$ and $\mathrm{O}$ are unit and null matrices of order 3 respectively, then

Let $a, b$ and $c$ be three real numbers satisfying
$\left[\begin{array}{lll} a & b & c \end{array}\right]\left[\begin{array}{lll} 1 & 9 & 7 \\ 8 & 2 & 7 \\ 7 & 3 & 7 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 0 \end{array}\right] \,\,\,(1)$

If the point $P(a, b, c)$ with reference to (1), lies on the plane $2 x+y+z=1$, then the value of $7 a+b+c$ is

Let $p$ be an odd prime number and $T_p$ be the following set of $2 \times 2$ matrices $T_p=\left\{A=\left[\begin{array}{ll}a & b \\ c & a\end{array}\right]: a, b, c \in\{0,1,2, \ldots,(p-1)\}\right\}$
The number of $A$ in $T_p$ such that $\det (A)$ is not divisible by $p$

If $P=\left[\begin{array}{rr}\cos \frac{\pi}{6} & \sin \frac{\pi}{6} \\ -\sin \frac{\pi}{6} & \cos \frac{\pi}{6}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P A P^T,$ then $P^T Q^{2007} P$ is equal to

If $A$ is a square matrix such that $A\;\left( {\mathrm{adj\;}A} \right) = \left[ {\begin{array}{cccccccccccccccccccc} 4&0&0\\ 0&4&0\\ 0&0&4 \end{array}} \right],$ then $\left| {\mathrm{adj}\;A} \right| =$

If $A = \left[ {\begin{array}{cccccccccccccccccccc} 1&{{{\log }_b}a}\\ {{{\log }_a}b}&1 \end{array}} \right]$, then $\left| A \right|$ is equal to

Let $A^2-A+I=O$, then inverse of $A$ is

If $A$ is a skew symmetric matrix of odd order, then its inverse is

If $X=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$, then the value of $X^n$ is

If $A=\left[\begin{array}{ll}0 & 3 \\ 2 & 0\end{array}\right]$ and $A^{-1}=\lambda(A)$, then $\lambda$

If $\omega$ is a non-real cube root of unity and $n$ is not a multiple of 3, then
$\Delta=\left|\begin{array}{ccc}1 & \omega^n & \omega^{2 n} \\ \omega^{2 n} & 1 & \omega^n \\ \omega^n & \omega^{2 n} & 1\end{array}\right|$ is equal to

The matrix $A=\left[\begin{array}{rrr}1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 & -4\end{array}\right]$ is nilpotent of index:

If matrix $A=\left[\begin{array}{ccc}1 & 0 & -1 \\ 3 & 4 & 5 \\ 0 & 6 & 7\end{array}\right] a$
and its inverse is denoted by
$\mathrm{A}^{-1}=\left[\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]$, then the value of $a_{23}=$

If $A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]$, then adj $\left(3 A^2+12 A\right)$ is equal to:

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?

If $A = \left[ {\begin{array}{cccccccccccccccccccc} {4\;\;\,2}\\ {3\;\;\,4} \end{array}} \right],\left| {\mathrm{adj}\;A} \right|$ is equal to

If $\alpha, \beta$, $\gamma$ are the roots of $x^3+a x^2+b=0,$ then the value of $\left|\begin{array}{lll}\alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta\end{array}\right|,$ is

Which of the following

(i) Adjoint of a symmetric matrix is symmetric

(ii) Adjoint of a unit matrix is a unit matrix

(iii) $\mathrm{A}(\operatorname{adj} \mathrm{A})=(\operatorname{adj} \mathrm{A}) \mathrm{A}=|\mathrm{A}| \mathrm{I}$ and

(iv) Adjoint of a diagonal matrix is a diagonal matrix, is/are incorrect