Matrices and Determinants

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is $-18$, then the value of the determinant of $A$ is ______

Let

$$M = \begin{bmatrix}\sin^4\theta & -1-\sin^2\theta\\ 1+\cos^2\theta & \cos^4\theta\end{bmatrix} = \alpha I + \beta M^{-1},$$

where $\alpha = \alpha(\theta)$ and $\beta = \beta(\theta)$ are real numbers, and $I$ is the $2\times2$ identity matrix. If

$\alpha^*$ is the minimum of the set $\{\alpha(\theta) : \theta\in[0,2\pi)\}$ and

$\beta^*$ is the minimum of the set $\{\beta(\theta) : \theta\in[0,2\pi)\}$,

then the value of $\alpha^* + \beta^*$ is

Let

$$M = \begin{bmatrix}0 & 1 & a\\ 1 & 2 & 3\\ 3 & b & 1\end{bmatrix}\quad\text{and}\quad \operatorname{adj}M = \begin{bmatrix}-1 & 1 & -1\\ 8 & -6 & 2\\ -5 & 3 & -1\end{bmatrix}$$

where $a$ and $b$ are real numbers. Which of the following options is/are correct?

Let $S$ be the sample space of all $3\times3$ matrices with entries from the set $\{0, 1\}$. Let the events $E_1$ and $E_2$ be given by

$E_1 = \{A\in S : \det A = 0\}$ and

$E_2 = \{A\in S : \text{sum of entries of } A \text{ is } 7\}$.

If a matrix is chosen at random from $S$, then the conditional probability $P(E_1|E_2)$ equals ______

Let

$$P_1 = I = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix},\quad P_2 = \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix},\quad P_3 = \begin{bmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix},$$

$$P_4 = \begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix},\quad P_5 = \begin{bmatrix}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix},\quad P_6 = \begin{bmatrix}0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{bmatrix}$$

and

$$X = \sum_{k=1}^6P_k\begin{bmatrix}2 & 1 & 3\\ 1 & 0 & 2\\ 3 & 2 & 1\end{bmatrix}P_k^T$$

where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?

Let $x\in\mathbb R$ and let

$$P = \begin{bmatrix}1 & 1 & 1\\ 0 & 2 & 2\\ 0 & 0 & 3\end{bmatrix},\quad Q = \begin{bmatrix}2 & x & x\\ 0 & 4 & 0\\ x & x & 6\end{bmatrix}\quad\text{and}\quad R = PQP^{-1}.$$

Then which of the following options is/are correct?

Suppose

$$\det\begin{bmatrix}\displaystyle\sum_{k=0}^nk & \displaystyle\sum_{k=0}^n{}^nC_k\,k^2\\ \displaystyle\sum_{k=0}^n{}^nC_k\,k & \displaystyle\sum_{k=0}^n{}^nC_k\,3^k\end{bmatrix} = 0$$

holds for some positive integer $n$. Then $\displaystyle\sum_{k=0}^n\frac{{}^nC_k}{k+1}$ equals ______

Let $S$ be the set of all column matrices $\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$ such that $b_1, b_2, b_3 \in \mathbb{R}$ and the system of equations (in real variables)

$$\begin{aligned} -x + 2y + 5z &= b_1 \\ 2x - 4y + 3z &= b_2 \\ x - 2y + 2z &= b_3 \end{aligned}$$

has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \in S$?

Let $P$ be a matrix of order $3 \times 3$ such that all the entries in $P$ are from the set $\{-1, 0, 1\}$. Then, the maximum possible value of the determinant of $P$ is _____ .

Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?

For a real number $\alpha$, if the system

$$\begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$$

of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 =$

How many $3 \times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there, for which the sum of the diagonal entries of $M^TM$ is 5?

If $f(x) = \begin{vmatrix} \cos(2x) & \cos(2x) & \sin(2x) \\ -\cos x & \cos x & -\sin x \\ \sin x & \sin x & \cos x \end{vmatrix}$, then

Let ω be a complex number such that $2\omega+1=z$ where $z=\sqrt{-3}$. If
$$\begin{vmatrix}1&1&1\\1&-\omega^2-1&\omega^2\\1&\omega^2&\omega^7\end{vmatrix}=3k,$$
then k is equal to :

If $A=\begin{bmatrix}2&-3\\-4&1\end{bmatrix}$, then adj $(3A^2+12A)$ is equal to :

If S is the set of distinct values of 'b' for which the following system of linear equations
$$x+y+z=1$$
$$x+ay+z=1$$
$$ax+by+z=0$$
has no solution, then S is :

The number of real values of λ for which the system of linear equations
$$2x+4y-\lambda z=0$$
$$4x+\lambda y+2z=0$$
$$\lambda x+2y+2z=0$$
has infinitely many solutions, is :

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

If $S=\left\{x\in[0,2\pi]:\begin{vmatrix}0&\cos x&-\sin x\\\sin x&0&\cos x\\\cos x&\sin x&0\end{vmatrix}=0\right\}$, then $\displaystyle\sum_{x\in S}\tan\left(\frac\pi3+x\right)$ is equal to :

For two 3 × 3 matrices A and B, let $A+B=2B'$ and $3A+2B=I_3$, where B′ is the transpose of B and $I_3$ is 3 × 3 identity matrix. Then :