Determinants Questions (2072)

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

Let $S$ denote the set of all real values of $\lambda$ such that the system $\lambda x+y+z=1$, $x+\lambda y+z=1$, $x+y+\lambda z=1$ is inconsistent. Then $\displaystyle\sum_{\lambda\in S}(|\lambda|^2+|\lambda|)$ is equal to:

$A = [a_{ij}]_{m \times n}$ is a square matrix, if

If $A=\dfrac{1}{5!6!7!}\begin{pmatrix}5!&6!&7!\\6!&7!&8!\\7!&8!&9!\end{pmatrix}$, then $|\text{adj}(\text{adj}(2A))|$ is equal to

$a+d=$

If det(\(A\)) = 5, then det(\(A^2\)) is:

The number of square matrices of order 5 with entries from the set $\{0, 1\}$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

If det(\(A\)) = 3, then \(\det(3A)\) for a 3 \times 3 matrix is:

Let $A$ be a $3 \times 3$ non-singular matrix then which of the following is not true

If the system of equations $x - 2y + 5z = 3$, $2x - y + z = 1$, and $11z - 7y + pz = q$ has infinitely many solutions, then

A square matrix $A$ of order 3 satisfies $A^2 = I - 2A$, where $I$ is an identity matrix of order 3. If $A^n = 29A - 12I$, then the value of $n$ is equal to

If determinant has one column as sum of two columns, then value is:

If determinant is skew-symmetric of odd order, its value is:

If det(\(A\)) = 2 and det(\(B\)) = 5, then \(\det(AB)\) is:

If $A$ is an idempotent matrix i.e. $A^2 = A$, and $B = I - A$, then which of the following is incorrect:

Let $A = [a_{rs}]$ be a $n \times n$ matrix such that $a_{rs} = (r-s)2^{(r-s)}$ where $i = \sqrt{-1}$, then $A = (\bar{A} \text{ denotes } [\bar{a}_{rs}] \text{ complex conjugate})$

If $A$ be $3 \times 3$ non-singular matrix, $|A| = K$, then $|(xA)^{-1}| = $ (where $x \neq 0$)

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

If \(A = [a_{ij}]_{n \times n}\) and \(a_{ij} = (i^2 + j^2 - ij)(j - i)\), where \(n\) is odd, then the value of \(tr.(A)\) is equal to:

Let $f(x)=\begin{vmatrix}\cos x&\cos^2x&\cos^4x\\\cos3x&\cos^23x&\cos^43x\\\cos5x&\cos^25x&\cos^45x\end{vmatrix}$, then $\displaystyle\int_0^{\pi}f(x)\,dx$ equals

For the system $\alpha x+y+z=1$, $x+\alpha y+z=1$, $x+y+\alpha z=\beta$, which statement is NOT correct?

If $M$ is a square matrix of order 3 such that $|M|=2$, then $\left|\text{adj}\!\left(\dfrac{M}{2}\right)\right|$ equals

If $1, \omega, \omega^2$ are cube roots of unity then system of equations: $x + 2\omega y + 3\omega^2 z = 1 - \omega^2$, $2x + 3\omega y + \omega^2 z = \omega^2 - \omega$, $3x + \omega y + 2\omega^2 z = \omega - 1$ has

If $A = \begin{vmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \end{vmatrix}$, $B = \text{adj}A$ and $C = 3A$, then $\frac{|\text{adj}(B)|}{|C|}$ is equal to

Let $A$, $B$ and $C$ be three $2\times2$ matrices with real entries such that $B=(I+A)^{-1}$ and $A+C=I$. If $BC=\begin{bmatrix}1&-5\\-1&2\end{bmatrix}$ and $CB\begin{bmatrix}x_1\\x_2\end{bmatrix}=\begin{bmatrix}12\\-6\end{bmatrix}$, then $x_1+x_2$ is

Let $A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}$, $d = |A| \neq 0$ and $|A - d(\text{Adj}A)| = 0$. Then

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)