Determinants Questions (2072)

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 2 & 2 & 0 \\ 2 & 2 & 2 \end{bmatrix}$, then $adj(adj A)$ is equal to -

Find the value of $\begin{vmatrix} 1 & 2 \\ -1 & 3 \end{vmatrix} \times \begin{vmatrix} 3 & 0 \\ -1 & 4 \end{vmatrix}$ and prove that it is equal to $\begin{vmatrix} 1 & 8 \\ -6 & 12 \end{vmatrix}$.

Given the system of linear equations\(2x + 2y + 3z = a\)\(3x - y + 5z = b\)\(x - 3y + 2z = c\)where \(a, b, c\) are non-zero real numbers, the system has more than one solution if

Let $P=\begin{bmatrix}0&2&\lambda\\2&3&1\\1&\mu&3\end{bmatrix}$ and $\text{Adj}(P)=\begin{bmatrix}10&-7&-1\\-5&-1&2\\-5&2&-4\end{bmatrix}$. Then $\left|(\text{adj}P)^{-1}+14\,\text{adj}(P^{-1})\right|$ equals

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

The determinant $$\begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix}$$ is equal to zero, if:

Let \lambda \in \mathbb{R}. The system of linear equations 2x_1 - 4x_2 + \lambda x_3 = 1, x_1 - 6x_2 + x_3 = 2, \lambda x_1 - 10x_2 + 4x_3 = 3 is inconsistent for

**Paragraph (continued):** Consider the system $x+2y-3z=a$; $2x+6y-11z=b$; $x-2y+7z=c$. Then the system: A) has a unique solution when $5a=2b+c$ B) has infinite number of solutions when $5a=2b+c$ C) has no solution for all $a,b,c$ D) has a unique solution for all $a,b,c$

If \(\det(A)\)=2, then det(A\)^3A^{-1}) equals:

If \(\det(A)\)=3, then det(A\)^4A⁻^4A^2) equals:

If \(\det(A)\)=5, then det(A\)^6A^6A^{-1}^1) equals:

If \(\det(A)\)=k, then det(A\)^n(A\)⁻^n + \(I)(A\)⁻^n - \(I)) equals:

If \(\det(A)\)=4, then det(A\)^{-1}\(A\)^{-1}\(A) equals:

If \(\det(A)\)=3, then det(A\)^4(I\) - \(A\)⁻^4)(I\) + \(A\)⁻^4)(I\) - \(A\)⁻^4)(I\) + \(A\)⁻^4)) equals:

If \(\det(A)\)=3, then det(A\)^{-1}\(A\)^{-1}\(A\)^{-1}) equals:

If \(\det(A)\)=3, then det(A\)^4(I\) - \(A\)⁻^4)(I\) + \(A\)⁻^4)(I\) - \(A\)⁻^4)) equals:

If \(\det(A)\)=2, then det(A\)^3(A\)⁻^3 + \(I)(A\)⁻^3 - \(I)(A\)⁻^3 + \(I)) equals:

If \(\det(A)\)=3, then det(A\)^4A^4A⁻^7) equals:

If determinant \(\Delta\) \(\neq\) 0, then rank of matrix is:

If \(\det(A)\)=k, then det(A\)^nA^mA⁻^m) where n>0 equals:

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$

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 $A = \frac{1}{3}\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ 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.