Determinants Questions (2072)

If $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is an orthogonal matrix of order 3, then:

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

If $A = \begin{vmatrix} \sin\theta\cos\theta & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi\sin\phi & \sin\theta\sin\phi & 0 \end{vmatrix}$ then

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

Matrix $A$ is such that $A^2 = 2A - I$, where $I$ is identity matrix, then for $n \geq 2, A^n =$

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

A be the set of all square matrices of order 3 with elements either 0, 1, or $-1$, then:

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 $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $P = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$, $Q = P^T AP$, then $PQ^{2014}P^T =$

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

Which of the following is true:

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 $a, b, c$ are the roots of the equation $x^3 + 2x^2 + 1 = 0$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} =$

For $3 \times 3$ Matrices $M$ and $N$, which of the following statement(s) is (are) not correct

Number of distinct real values of $K$, such that the system of equations $x+2y+z=1$, $x+3y+4z=K$, $x+5y+10z=K^2$ has infinitely many solutions is:

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?

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:

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

If $f(x) = \begin{vmatrix} x & 1+x^2 & x^3 \\ \ln(1+x^2) & e^x & \sin x \\ \cos x & \tan x & \sin^2 x \end{vmatrix}$, then:

Let $f(x)=\displaystyle\int\frac{7x^{10}+9x^8}{(1+x^2+2x^9)^2}dx$, $x>0$, $\lim_{x\to0}f(x)=0$ and $f(1)=\dfrac{1}{4}$. If $A=\begin{bmatrix}0&0&1\\\frac{1}{4}&f'(1)&1\\\alpha^2&4&1\end{bmatrix}$ and $B=\text{adj}(\text{adj}\,A)$ be such that $|B|=81$, then $\alpha^2$ is equal to

Let $A$ and $B$ are square matrices of order 2 such that $A + adj(B^T) = \begin{bmatrix} 3 & 2 \\ 2 & 3 \end{bmatrix}$ and $adj(A^T) - adj(B) = \begin{bmatrix} -2 & -1 \\ -1 & -2 \end{bmatrix}$, then $A + 2A^T + 3A^T$ is

For the system $2x-y+3z=5$, $3x+2y-z=7$, $4x+5y+\alpha z=\beta$, which of the following is NOT correct?

For the system $2x+4y+2az=b$, $x+2y+3z=4$, $2x+5y+2z=8$, which of the following is NOT correct?

Which relation is incorrect for determinant \Delta with cofactors?

Matrix $A$ is such that $A^2 = 2A - I$, where $I$ is identity matrix, then for $n \geq 2, A^n =$

Prove that $\begin{vmatrix} a_1x_1 + b_1y_1 & a_1x_2 + b_1y_2 & a_1x_3 + b_1y_3 \\ a_2x_1 + b_2y_1 & a_2x_2 + b_2y_2 & a_2x_3 + b_2y_3 \\ a_3x_1 + b_3y_1 & a_3x_2 + b_3y_2 & a_3x_3 + b_3y_3 \end{vmatrix} = 0$

If det(\(A\)) = 3, then det(\(A\)⁻^1) is:

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

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