Matrices & Determinants Questions (2045)

Let \(\omega\) be a complex number such that \(2\omega + 1 = z\) where \(z = \sqrt{-3}\). If\[\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2-1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k,\]then k is equal to

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

94. Let A and B are square matrices of same order satisfying \(AB = A\) and \(BA = B\), then \((A^{2019} + B^{2019})^{2020}\) is equal to:

996. Let \(P\) be a \(2 \times 2\) matrix such that \(P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}\) and \(P^2\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}\). If \(x_1\) and \(x_2\) are two values of \(x\) for which \(|P - xI| = 0\), where \(I\) is an identity matrix of order 2, then find the value of \(x_1^2 + x_2^2\).

Sum of elements of (adjA) B is -

Given: \(x + ay + z = 3,\; x + 2y + 2z = 6,\; x + 5y + 3z = b\).For no solution, which of the following is correct?

The set of equations \(\lambda x - y + (\cos\theta)z = 0\), \(3x + y + 2z = 0\), \((\cos\theta)x + y + 2z = 0\), \(0 \leq \theta

Let $X = (ABA^T)^{2020}$ where $B$ is symmetric. If $X^T = ((ABA^T)^{mm})^2 - ((ABA^T)^{2020}) - (ABA^T)^{2020}$ and $X^T = X$ is a symmetric matrix, then $a_{21} = b_{13} = c_1$, $b_0 = c_2$. Required value = ?

If \det(A)=k, then \det(c\(A\)^T) for \(n \times n\) matrix equals:

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

If det(\(A\)) = -4, then det(\(A^T\)) is:

Let \[\Delta = \begin{vmatrix} x^2+x & x+1 & x-2 \\ 2x^2+3x-1 & 3x & 3x-3 \\ x^2+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = ax - 12\]Then the value of \(a\) is:

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

Given \(a = x/(y-z)\), \(b = y/(z-x)\), and \(c = z/(x-y)\), where \(x, y\) and \(z\) are not all zero, then the value of \(ab + bc + ca\) is

Question 87: Statement-1: The value of the determinant $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix} = 2$.Statement-2: Neither of two rows or columns of [a matrix property regarding determinants].

The system of equations x + y + z = 5; x + 2y + 3z = 9; x + 3y + Dz = I is called lazy, if it has no solution. The condition for this is

If \(A\) is a \(3 \times 3\) matrix with \det(A)=a, then \det(adj(adj \(A)) equals:

Evaluate the determinant $\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ bc & ac & ab \end{vmatrix}$.

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

Let $A = [a_{ij}]_{n \times n}$ where $a_{ij} = i^2 - j^2$. Then $A$ is

Let a, b, and c be such that b(a + c) ≠ 0. If\[\begin{vmatrix} a & a+1 & a-1 \\ -b & b+1 & b-1 \\ c & c-1 & c+1 \end{vmatrix} + \begin{vmatrix} a+1 & b+1 & c-1 \\ a-1 & b-1 & c+1 \\ (-1)^{n+2}a & (-1)^{n+1}b & (-1)^n c \end{vmatrix} = 0,\] then the value of n is

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

Consider a matrix A(θ) = sinθcosθ-cosθsinθ then

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

67. If \(x, y, z, p, q, r, l, m\) and \(n\) are in geometric progression, find the value of the determinant \[\begin{vmatrix} \log x & \log y & \log z \\ \log p & \log q & \log r \\ \log l & \log m & \log n \end{vmatrix}.\]

The system of linear equations x + λy - z = 0, λx - y - z = 0, x + y - λz = 0 has a non-trivial solution for :

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:

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

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

$\frac{c}{b}=$

Which of the following is true:

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

If $x^a y^b = e^m$, $x^c y^d = e^n$, $P = \begin{pmatrix} m & b \\ n & d \end{pmatrix}$, $Q = \begin{pmatrix} a & m \\ c & n \end{pmatrix}$, $R = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, then:

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

If $A(\theta) = \begin{pmatrix} \sin \theta & i \cos \theta \\ i \cos \theta & \sin \theta \end{pmatrix}$, then which of the following is not true

If $\begin{vmatrix} bc-a^2 & ca-b^2 & ab-c^2 \\ ca-b^2 & ab-c^2 & bc-a^2 \\ ab-c^2 & bc-a^2 & ca-b^2 \end{vmatrix} = \begin{vmatrix} a^2 & b^2 & b^2 \\ b^2 & a^2 & b^2 \\ b^2 & b^2 & a^2 \end{vmatrix}$, then

$\Delta = \begin{vmatrix} a & a^2 & 0 \\ 1 & 2a+b & a+b \\ 0 & 1 & 2a+3b \end{vmatrix}$ is divisible by

Which of the following values of $a$ satisfy the equation $\begin{vmatrix} (1+a)^2 & (1+2a)^2 & (1+3a)^2 \\ (2+a)^2 & (2+2a)^2 & (2+3a)^2 \\ (3+a)^2 & (3+2a)^2 & (3+3a)^2 \end{vmatrix} = -684a$

The equations \((\lambda - 1)x + (3\lambda + 1)y + 2\lambda z = 0\), \((\lambda - 1)x + (4\lambda - 2)y + (\lambda + 3)z = 0\) and \(2x + (3\lambda + 1)y + 3(\lambda - 1)z = 0\) give non-trivial solution for some values of \(\lambda\). Then the ratio \(x : y : z\), when \(\lambda\) has smallest of these values 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 \(\det(A)\)=k, then det(A\)^nA^nA^n) equals:

The value of $\theta$ lying between $\theta = 0$ and $\theta = \frac{\pi}{2}$ and satisfying the equation $$\begin{vmatrix} 1+\sin^2\theta & \cos^2\theta & 4\sin 4\theta \\ \sin^2\theta & 1+\cos^2\theta & 4\sin 4\theta \\ \sin^2\theta & \cos^2\theta & 1+4\sin 4\theta \end{vmatrix} = 0$$ is:

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero symmetric matrix. Then which of the following is (are) skew-symmetric:

If $a, b, c$ are non-zero real numbers such that $$\begin{vmatrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \end{vmatrix} = 0$$, then:

Let $\{\Delta_1, \Delta_2, \Delta_3, \ldots \Delta_k\}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_1, a_2, \ldots a_9$. Then:

For α, β ∈ R and a natural number n, let A_r = r1n22+α2r2n2-β3r-23n(3n-1)2. Then 2A_10 - A_8 is

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

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

$a+d=$

$\frac{c}{b}=$