Matrices and Determinants

If $x=a$, $y=b$, $z=c$ is a solution of the system of linear equations
$$x+8y+7z=0$$
$$9x+2y+3z=0$$
$$x+y+z=0$$
such that the point (a, b, c) lies on the plane $x+2y+z=6$, then $2a+b+c$ equals :

Let $P=\begin{bmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{bmatrix}$, where $\alpha\in\mathbb R$. Suppose $Q=[q_{ij}]$ is a matrix such that $PQ=kI$, where $k\in\mathbb R$, $k\ne0$ and $I$ is the identity matrix of order 3. If $q_{23}=-\dfrac k8$ and $\det(Q)=\dfrac{k^2}{2}$, then

The total number of distinct $x\in\mathbb R$ for which $\begin{vmatrix}x&x^2&1+x^3\\2x&4x^2&1+8x^3\\3x&9x^2&1+27x^3\end{vmatrix}=10$ is

Let $z=\dfrac{-1+\sqrt3\,i}{2}$, where $i=\sqrt{-1}$, and $r,s\in\{1,2,3\}$. Let $P=\begin{bmatrix}(-z)^r&z^{2s}\\z^{2s}&z^r\end{bmatrix}$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2=-I$ is

Let $P=\begin{bmatrix}1&0&0\\4&1&0\\16&4&1\end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q=[q_{ij}]$ is a matrix such that $P^{50}-Q=I$, then $\dfrac{q_{31}+q_{32}}{q_{21}}$ equals

Let $a,\lambda,\mu\in\mathbb R$. Consider the system of linear equations
$$ax+2y=\lambda$$
$$3x-2y=\mu$$
Which of the following statement(s) is(are) correct?

If $A=\begin{bmatrix}5a&-b\\3&2\end{bmatrix}$ and $A\operatorname{adj}A=AA^T$, then $5a+b$ is equal to :

The system of linear equations
$$x+\lambda y-z=0$$
$$\lambda x-y-z=0$$
$$x+y-\lambda z=0$$
has a non-trivial solution for :

If $P=\begin{bmatrix}\frac{\sqrt3}2&\frac12\\-\frac12&\frac{\sqrt3}2\end{bmatrix}$, $A=\begin{bmatrix}1&1\\0&1\end{bmatrix}$ and $Q=PAP^T$, then $P^TQ^{2015}P$ is :

The number of distinct real roots of the equation, $\begin{vmatrix}\cos x&\sin x&\sin x\\\sin x&\cos x&\sin x\\\sin x&\sin x&\cos x\end{vmatrix}=0$ in the interval $\left[-\dfrac\pi4,\dfrac\pi4\right]$ is :

Let A be a 3 × 3 matrix such that $A^2-5A+7I=O$.

Statement - I : $A^{-1}=\dfrac17(5I-A)$.

Statement - II : The polynomial $A^3-2A^2-3A+I$ can be reduced to 5(A − 4I).

Then :

If $A=\begin{bmatrix}-4&-1\\3&1\end{bmatrix}$, then the determinant of the matrix $\left(A^{2016}-2A^{2015}-A^{2014}\right)$ is :

The number of distinct real values of λ for which the lines $\dfrac{x-1}1=\dfrac{y-2}2=\dfrac{z+3}{\lambda^2}$ and $\dfrac{x-3}1=\dfrac{y-2}{\lambda^2}=\dfrac{z-1}2$ are coplanar is :

Let $X$ and $Y$ be two arbitrary, $3\times3$, non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3\times3$, non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

Which of the following values of $\alpha$ satisfy the equation
$$\begin{vmatrix}(1+\alpha)^2&(1+2\alpha)^2&(1+3\alpha)^2\\(2+\alpha)^2&(2+2\alpha)^2&(2+3\alpha)^2\\(3+\alpha)^2&(3+2\alpha)^2&(3+3\alpha)^2\end{vmatrix}=-648\alpha\ ?$$

If $A=\begin{bmatrix}1&2&2\\2&1&-2\\a&2&b\end{bmatrix}$ is a matrix satisfying the equation $AA^T=9I$, where I is 3×3 identity matrix, then the ordered pair (a, b) is equal to :

The set of all values of λ for which the system of linear equations :
$$2x_1-2x_2+x_3=\lambda x_1$$
$$2x_1-3x_2+2x_3=\lambda x_2$$
$$-x_1+2x_2=\lambda x_3$$
has a non-trivial solution,

The least value of the product $xyz$ for which the determinant $\begin{vmatrix}x&1&1\\1&y&1\\1&1&z\end{vmatrix}$ is non-negative, is :

If $A=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$, then which one of the following statements is not correct ?

If A is a 3×3 matrix such that $|5\cdot\text{adj}A|=5$, then $|A|$ is equal to :