Matrices and Determinants

Let the matrix $A=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$ and the matrix $B_0=A^{49}+2A^{98}$. If $B_n=\operatorname{Adj}(B_{n-1})$ for all $n\ge1$, then $\det(B_4)$ is equal to :

Let $A=\begin{bmatrix}1&-1\\2&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}\beta&1\\1&0\end{bmatrix}$, $\alpha,\beta\in\mathbf R$. Let $\alpha_1$ be the value of $\alpha$ which satisfies $(A+B)^2=A^2+\begin{bmatrix}2&2\\2&2\end{bmatrix}$ and $\alpha_2$ be the value of $\alpha$ which satisfies $(A+B)^2=B^2$. Then $|\alpha_1-\alpha_2|$ is equal to ______.

Let A and B be any two $3\times3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let A and B be two 3 × 3 non-zero real matrices such that AB is a zero matrix. Then

The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

Let p and p + 2 be prime numbers and let

$$\Delta=\begin{vmatrix}p! & (p+1)! & (p+2)!\\ (p+1)! & (p+2)! & (p+3)!\\ (p+2)! & (p+3)! & (p+4)!\end{vmatrix}$$

Then the sum of the maximum values of α and β, such that $p^\alpha$ and $(p+2)^\beta$ divide Δ, is _______.

Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix}-1&2\\1&-1\end{bmatrix}$ by a single elementary row operation ?

If the system of equations

$x+y+z=6$
$2x+5y+\alpha z=\beta$
$x+2y+3z=14$

has infinitely many solutions, then α + β is equal to

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is :

Let $X=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $A=\begin{bmatrix}-1&2&3\\0&1&6\\0&0&-1\end{bmatrix}$. For k ∈ ℕ, if $X'A^kX=33$, then k is equal to _______.

If the system of linear equations

$\lambda x+y-z=-1$
$x-y-3z=2$
$-x+y+z=\mu$

has infinitely many solutions, then the equation of the line passing through the points $(\lambda+2\mu,\ 2\lambda+\mu)$ and $(1,\ \lambda\mu)$ is

Let A be a matrix and $\begin{bmatrix}1&2\\0&1\end{bmatrix}A\begin{bmatrix}2&0\\-1&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. If |A| and |A$^2$| are the roots of the quadratic equation $ax^2+bx+3=0$, then a + b − ab is equal to

The value of $|M|$ is ____.

The value of $D$ is ____.

For any $3\times3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let
$$E = \begin{bmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 8 & 13 & 18\end{bmatrix},\quad P = \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}\quad\text{and}\quad F = \begin{bmatrix}1 & 3 & 2\\ 8 & 18 & 13\\ 2 & 4 & 3\end{bmatrix}$$
If $Q$ is a nonsingular matrix of order $3\times3$, then which of the following statements is (are) TRUE?

For any $3\times3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3\times3$ identity matrix. Let $E$ and $F$ be two $3\times3$ matrices such that $(I - EF)$ is invertible. If $G = (I - EF)^{-1}$, then which of the following statements is (are) TRUE?

The system of linear equations

$3x-2y-kz=10$
$2x-4y-2z=6$
$x+2y-z=5m$

is inconsistent if :

Let $P=\begin{bmatrix}3&-1&-2\\2&0&\alpha\\3&-5&0\end{bmatrix}$, where α ∈ ℝ. Suppose Q = [q$_{ij}$] is a matrix satisfying PQ = kI$_3$ for some non-zero k ∈ ℝ. If $q_{23}=-\dfrac k8$ and $|Q|=\dfrac{k^2}2$, then α$^2$ + k$^2$ is equal to _______.

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M$^T$M is seven, is _______.

Let $M$ be a $3\times3$ invertible matrix with real entries and let $I$ denote the $3\times3$ identity matrix. If $M^{-1} = \operatorname{adj}(\operatorname{adj} M)$, then which of the following statements is/are ALWAYS TRUE?