Determinants Questions (2072)

If \(\Delta\) = 5 and one row is multiplied by 3, new determinant is:

If \(\det(A)\)=4, then det(A\)^5(A\)⁻^5 - \(I)) equals:

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

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 \(x_1^2 + x_2^2\).

If \(\det(A)\)=5, then det(A\)^6(A\)⁻^6 + \(I)(A\)⁻^6 - \(I)) equals:

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:

$a+d=$

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 $PQ^{2014}P^T =$

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

The ratio of the trace of matrix B to matrix C is (where C = (A ⋅ B))

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

If $A$ is a square matrix of order $n \times n$ and $k$ is a scalar, then $adj(kA)$ is equal to

If \(A = [a_{ij}]_{4 \times 4}\), such that \(a_{ij} = \begin{cases} 2, & \text{when } i = j \\ 0, & \text{when } i \neq j \end{cases}\), then \(\left\{ \dfrac{\det(\text{adj}(\text{adj } A))}{7} \right\}\) is (where \(\{\cdot\}\) represents fractional part function)

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

If $A, B$ are two matrices such that $A + B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}, A - B = \begin{bmatrix} 3 & 2 \\ -2 & 0 \end{bmatrix}$, then find $AB$.

For the system $x+y+z=6$, $\alpha x+\beta y+7z=3$, $x+2y+3z=14$, which of the following is NOT true?

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

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:

If $ax + by + cz = 0$, $bx + cy + az = 0$, $cx + ay + bz = 0$, $a,b,c \in \mathbb{R}^+$ then:

Consider the system $\alpha x+2y+z=1$, $2\alpha x+3y+z=1$, $3x+\alpha y+2z=\beta$, $\alpha,\beta\in\mathbb{R}$. Which of the following is NOT correct?

The system of equations $6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0$ has:

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

Let An and Bn be square matrices of order 3, which are defined as:An = [aij] and Bn = [bij] where \(a_{ij} = \frac{2i+j}{3 \cdot 2^n}\) and \(b_{ij} = \frac{3i-j}{2^n}\) for all i and j, \(1 \leq i, j \leq 3\).If \(\lambda = \lim_{n \to \infty} \text{Tr.}(3A_1 + 3^2A_2 + 3^3A_3 + \cdots + 3^nA_n)\) and\(\mu = \lim_{n \to \infty} \text{Tr.}(2B_1 + 2^2B_2 + 2^3B_3 + \cdots + 2^nB_n)\), then find the value of \(\frac{\lambda + \mu}{3}\)[Note: Tr.(P) denotes the trace of matrix P.]

If \(\det(A)\)=k, then \(\det(I)\) equals:

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

Let the system $x+y+kz=2$, $2x+3y-z=1$, $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$, $(2k+1)x+(k+5)y=10$ has:

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:

If \(S = \left\{ x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0 \right\}\), then \(\displaystyle\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)\) is equal to

Let A be the set of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ c & a \end{pmatrix}$, such that $a,b,c \in \{0,1,2,3,4\}$ then:

$A = \begin{pmatrix} -3 & -1 & 2 \\ 3 & 1 & -1 \\ 4 & 2 & 5 \end{pmatrix}$, $A \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $B = \begin{pmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{pmatrix}$, then:

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:

If $a, b$ and $c$ are distinct positive real numbers such that $\Delta_1 = \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$ and $\Delta_3 = \begin{vmatrix} bc - a^2 & ac - b^2 & ab - c^2 \\ ac - b^2 & ab - c^2 & bc - a^2 \\ ab - c^2 & bc - a^2 & ac - b^2 \end{vmatrix}$, then

Eigen values of matrix $\begin{bmatrix} 2 & 1 & 1 \\ 2 & 3 & 4 \\ -1 & -1 & -2 \end{bmatrix}$ are:

Let A be the set of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ c & a \end{pmatrix}$, such that $a,b,c \in \{0,1,2,3,4\}$ then:

If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then which of the following is true?

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

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:

$A = \begin{pmatrix} -3 & -1 & 2 \\ 3 & 1 & -1 \\ 4 & 2 & 5 \end{pmatrix}$, $A \begin{pmatrix} x_1 \\ y_1 \\ z_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_2 \\ y_2 \\ z_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$, $A \begin{pmatrix} x_3 \\ y_3 \\ z_3 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, $B = \begin{pmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ z_1 & z_2 & z_3 \end{pmatrix}$, then:

Which of the following is not the root of the equation \(\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix} = 0\)?

Find the value of \(x\) for which the matrix \[A = \begin{bmatrix} 2/x & -1 & 2 \\ 1 & x & 2x^2 \\ 1 & 1/x & 2 \end{bmatrix}\] is singular.

Let $A = \begin{bmatrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2p & 8 + 3p + 2q \\ 6 & 12 + 3p & 20 + 6p + 3q \end{bmatrix}$. If $\det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n$, where $m, n \in \mathbb{N}$, then $m + n$ is equal to

Let $A = \begin{bmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{bmatrix}$. If $AA^T = I^3$, then $|p|$ is :

Let $A = \begin{pmatrix}1&1\\1&1\end{pmatrix}$. If $\det(A^n - I) = 1 - \lambda^n$, find $\lambda$.

If $A = \begin{bmatrix} 1 & 3 & 5 \\ 3 & 5 & 1 \\ 5 & 1 & 3 \end{bmatrix}$, then $\text{adj } A$ is equal to -

Let $A=\begin{pmatrix}1&0&0\\\sqrt{a}&1&0\\a\sqrt{a}&\sqrt{b}&1\end{pmatrix}$; $a,b\in\mathbb{R}^+$. If for some $n\in\mathbb{N}$, $A^n=\begin{pmatrix}1&0&0\\72&1&0\\3600&72&1\end{pmatrix}$, then the number of triangles formed by joining the vertices of an $n$-sided polygon having no side common with the polygon is

Let $\begin{vmatrix}a&\sqrt{5}&\sqrt{7}\\\sqrt{3}&b&\sqrt{7}\\\sqrt{3}&\sqrt{5}&c\end{vmatrix}=0$, ($a\neq\sqrt{3}, b\neq\sqrt{5}, c\neq\sqrt{7}$) and $\dfrac{a}{a-\sqrt{3}}+\dfrac{b}{b-\sqrt{5}}+\dfrac{c}{c-\sqrt{7}}=\lambda$. If $a=2\sqrt{3}$, then the point $(b^2,c^2)$ may lie on the line

The set of natural numbers is divided into arrays of rows and columns in the form of matrices as $A_1=[1]$, $A_2=\begin{bmatrix}2&3\\4&5\end{bmatrix}$, $A_3=\begin{bmatrix}6&7&8\\9&10&11\\12&13&14\end{bmatrix}$ and so on. Let the trace of $A_{10}$ be $\lambda$. Find unit digit of $\lambda$.

If $\begin{vmatrix}1+x&1&1\\1&1+y&1\\1&1&1+z\end{vmatrix}=0$, then the value of $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}$ is

If $A$ is a $3\times3$ skew-symmetric matrix, find $\det(2A+I)$ given $\det(A+I)=2$

Let $A = \begin{pmatrix}1&1\\1&1\end{pmatrix}$. If $\det(A^n - I) = 1 - \lambda^n$, find $\lambda$.