Matrices & Determinants Questions (2045)

Let $P=[p_{ij}]$ and $Q=[q_{ij}]$ be two square matrices of order 3 such that $q_{ij}=2^{(i+j-1)}p_{ij}$ and $\det(Q)=2^{10}$. Then the value of $\det(\text{adj}(\text{adj}\,P))$ is:

If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2$, then the ordered pair $(A, B)$ is equal to

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) $A^{26}C^{13} - C^{13}A^{26}$ is symmetric. Then,

If $A=\begin{pmatrix}1&5\\\lambda&10\end{pmatrix}$, $A^{-1}=\alpha A+\beta I$ and $\alpha+\beta=-2$, then $4\alpha^2+\beta^2+\lambda^2$ is equal to:

The system of homogeneous equations $\lambda x + (\lambda + 1)y + (\lambda - 1)z = 0$, $(\lambda + 1)x + \lambda y + (\lambda + 2)z = 0$, $(\lambda - 1)x + (\lambda + 2)y + \lambda z = 0$ has non trivial solution for:

$A_1$ is a matrix formed by replacing all the elements in $A_{n \times n}$ by corresponding cofactors. $A_2$ is matrix formed by replacing all elements of $A_1$ by corresponding cofactors and $A_3, A_4 \ldots$ formed so on. if $|A_n| = K$, then $|A| =$

If determinant has repeated elements across rows, then first step should be:

If \(\Delta\) = \[ \begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{vmatrix} \], then \(\Delta\) equals:

Let $A = \begin{pmatrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -i \\ 0 & 1 \end{pmatrix}$, where $i = \sqrt{-1}$. If $M = A^T B A$, then the inverse of the matrix $AM^{2023}A^T$ is

Let $S_1$ and $S_2$ be the sets of all $a\in\mathbb{R}-\{0\}$ for which the system $ax+2ay-3az=1$, $(2a+1)x+(2a+3)y+(a+1)z=2$, $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions respectively. Then:

If determinant contains variables in one row only, best strategy is:

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

Let for $A=\begin{pmatrix}1&2&3\\\alpha&3&1\\1&1&2\end{pmatrix}$, $|A|=2$. If $|2\,\text{adj}(2\,\text{adj}(2A))|=32^n$, then $3n+\alpha$ is equal to

The number of positive integral solutions of the equation $\begin{vmatrix} x^2+1 & x y & x z^2 \\ x y & y^2+1 & y z^2 \\ x z^2 & y z^2 & z^2+1 \end{vmatrix} = 11$ is are

The set of all values of $t \in \mathbb{R}$, for which the matrix $\begin{pmatrix} e^t & e^{-t}(\sin t - 2\cos t) & e^{-t}(-2\sin t - \cos t) \\ e^t & e^{-t}(2\sin t + \cos t) & e^{-t}(\sin t - 2\cos t) \\ e^t & e^{-t}\cos t & e^{-t}\sin t \end{pmatrix}$ is invertible, is

Consider a skew-symmetric matrix $A = \begin{bmatrix} 0 & b & k \\ b & 0 & c \\ -c & -c & 0 \end{bmatrix}$ such that $a, b$ and $c$ are selected from the set $S = \{0, 1, 2, 3, \ldots, 12\}$. If $|A|$ is divisible by 3, then the number of such possible matrices is

Let $S$ be the set of all $3 \times 3$ symmetric matrices whose entries are either $0$ or $1$. Two of these entries are $1$ and four of them are $0$. A matrix is selected from set $S$, what is the probability that the selected matrix is non singular

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

If \Delta = \[ \begin{vmatrix} a1 & b1 & c1 \\ & a2 & b2 & c2 \\ & a3 & b3 & c3 \end{vmatrix} \] and A2, B2, C2 are cofactors of second row, then a1A2 + b1B2 + c1C2 equals:

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

Which of the following is true:

If $A, B$ are two non-singular matrices of order $3$ and $I$ is an identity matrix of order $3$ such that $AA^T = 5I$ and $3A^{-1} = 2A^T$ - $\operatorname{adj}(AH)$, then $|B^n|$ is equal to

Which of the following matrices do not have eigen values 1 and $-1$

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix where $a_{ij} = \begin{cases} (i^2 - j^2 + 2)z & i < j \\ i & i \geq j \\ 0 & i = j \end{cases}$, then the minimum value of $|A|$ is equal to (where $z$ is a real number)

If $A^2 = A$, then $(A + I)^5$ is equal to (where $I$ is identity matrix)

If det(\(A\)) = 4 and det(\(B\)) = 2, then det(\(A^2\)B) is:

If determinant has symmetric entries, then best approach is:

Let $f(x) = \begin{vmatrix} x \cos x & 2x \sin x & x \tan x \\ 1 & 2x & 1 \end{vmatrix}$, then $\lim_{x \to 0} \frac{f(x)}{x^2} =$

Let $A = [a_{ij}]_{3 \times 3}$, $B = [b_{ij}]_{3 \times 3}$ where $b_{ij} = 3^{i-j}a_{ij}$, $C = [c_{ij}]_{3 \times 3}$, where $c_{ij} = 4^{i-j}b_{ij}$ be any three matrices. If $|A| = 4$, then $|B| + |C| = (|X|$ denotes determinant of matrix $X)$

Let \(A\) be any \(3 \times 3\) invertible matrix. Then, which one of the following is not always true?

If \(\Delta\) = \[ \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix} \], then \(\Delta\) equals:

If all elements of a row are multiplied by k, determinant becomes:

If \(\Delta\) = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and a = d = g, then \(\Delta\) equals:

If determinant has a row with two identical elements, best approach is:

If determinant has two columns proportional, then value is:

If \begin{vmatrix} (x+1) & (x+1)^2 & (x+1)^3 \\ (x+2) & (x+2)^2 & (x+2)^3 \\ (x+3) & (x+3)^2 & (x+3)^3 \end{vmatrix} is expressed as a polynomial in \(x\), then the term independent of \(x\) is:

If the matrix \(\begin{pmatrix} 0.3 & b & c \\ l & m & n \\ 0 & p & q \end{pmatrix}\) is an orthogonal matrix, find the sum of all possible values of \(10(mq - np)\).

If the system of linear equations\(x + 2ay + az = 0\)\(x + 3by + bz = 0\)\(x + 4cy + cz = 0\)has a non-zero solution, then \(a\), \(b\), \(c\) are in:

If determinant has a common factor in a row, then determinant becomes:

The value of \((3 \; 2 \; 0) U^{-1} \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}\) is

If determinant is such that two rows differ only by a constant factor, then determinant is:

For the matrix $A = \begin{bmatrix} 4 & -4 & 5 \\ -2 & 3 & -3 \\ 3 & -3 & 4 \end{bmatrix}$ find $A^{-2}$.

If the system of equations $x+2ay+az=0$, $x+3by+bz=0$, $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$

Let \(A + B = \begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}\) where \(A\) is a symmetric matrix and \(B\) is a skew-symmetric matrix. Find \(AB\).

Given A2 − A + I = O, then A−1 equals:

If the system of equations \(x = 2a\), \(y = 3b\), \(z = c\) has non-zero solutions, i.e., \(\Delta = 0\) where \(\Delta = \begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{vmatrix} = 0\), then \(a\), \(b\), \(c\) are in:

If $\begin{vmatrix}\sin x+1&\sin 2x&\sin 3x\\ \sin 2x&\sin 3x+a&\sin 4x\\ \sin 3x&\sin 4x&\sin 5x+a^2\end{vmatrix}=2025(f(x)+45)$ where $f(x)$ is a function of $x$ and $a$ is complex, then sum of all possible values of $a$ is

Let $A$ be a non-singular idempotent matrix of order $2025\times2025$. Consider statements: (i) Trace of $A$ = 2025, (ii) $A$ has to be a scalar matrix, (iii) Trace of adjoint of $A^2$ = 2025. Which are true?

If \(a_1, a_2, a_3, \ldots, a_n, \ldots\) are in GP, then the value of the determinant \[\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}\] is

If \(D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}\), for \(x \neq 0,\, y \neq 0\) then \(D\) is