Matrices & Determinants Questions (2045)

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

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

Let $A$ be a $3\times 3$ matrix of non-negative real numbers such that $A\begin{bmatrix}2\\2\\2\end{bmatrix}=4\begin{bmatrix}1\\1\\1\end{bmatrix}$. Then $(\det A)_{\max}$ is

$\text{tr}(A)$ is equal to

If \(S\) is the set of distinct values of \(b\) for which the following system of linear equations:\(x + y + z = 1\)\(x + ay + z = 1\)\(ax + by + z = 0\)has no solution, then \(S\) is

For what values of \(x\): \([1\quad 2\quad 1]\begin{bmatrix}1 & 2 & 0\\2 & 0 & 1\\1 & 0 & 2\end{bmatrix}\begin{bmatrix}0\\2\\x\end{bmatrix} = 0\,?

If $A = \begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then adj $A =$

If the system of equations $x+2y+3z=3$, $4x+3y-4z=4$, $8x+4y-\lambda z=9+\mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to:

Let $A = \begin{pmatrix}1&2&2\\2&1&1\\2&2&1\end{pmatrix}$. If $A$ is a zero divisor of $x^2 - 4x - 5$, find $\text{Tr}(A^3)$.

In the matrix $A = \begin{bmatrix} 2 & 5 & 19 & 0 \\ 1 & 2 & 0 & 1 \\ 2 & 7 & \sqrt{3} & \sqrt{5} \end{bmatrix}$(i) The order of the matrix,(ii) The number of elements,(iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$.

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

If $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$, then $A^5 =$

If $A = \begin{bmatrix} -2 & -1 & 1 \\ -1 & 7 & 4 \\ 1 & -x & -3 \end{bmatrix}$ be symmetric matrix then find the value of $x$.

If $A$ and $B$ are matrices of order $m \times n$ and $n \times m$ respectively, then order of matrix $B^T(A^T)^T$ is -

Obtain the inverse of the following matrix using elementary operations $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$.

Let $\alpha$ be a solution of $x^2 + x + 1 = 0$, and for some $a$ and $b$ in $\mathbb{R}$, $$\begin{bmatrix} 4 & a & b \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 2 & -2 \\ -14 & -8 \end{bmatrix} = \begin{bmatrix} 0 & 0 \end{bmatrix}$$ If $\frac{4}{\alpha^2} + \frac{m}{\alpha} + n = 3$, then $m + n$ is equal to

70. If \(y = \sin(mx)\) and \(y_n = \dfrac{d^n y}{dx^n}\), then the determinant \[\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix} =\] ______.

Let $a \in \mathbb{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\det(A) = -4$ and $A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}$, where $I$ is the identity matrix of order $3 \times 3$. If $\det((a+1)\operatorname{adj}((a-1)A)) = \frac{2^m}{3^n}$, $m, n \in \{0, 1, 2, \ldots, 20\}$, then $m + n$ is equal to:

If $A = \begin{bmatrix} 0 & -1 & 2 \\ 2 & -2 & 0 \end{bmatrix}$, $B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $M = AB$, then $M^{-1}$ is equal to

Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = \begin{bmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{bmatrix}$, $|A| = -1$. Let $B$ be the inverse of the matrix $\text{adj}(A \cdot \text{adj}(A))$. Then $|\lambda B + I|$ is equal to ______

Let $A$ be a $3 \times 3$ matrix such that $|\text{adj}(\text{adj}(\text{adj}A))| = 81$. If $S = \{n \in \mathbb{Z} : (|\text{adj}(\text{adj}A)|)^{2(n-1)} = |A|^{2(3n^2-5n-4)}\}$, then $\sum_{n \in S} |n^2 + n|$ is equal to

If all rows multiplied by k, determinant becomes:

If \(A\) is identity matrix, then det(A\)^{-1}) equals:

The number of singular matrices of order 2 , whose elements are from the set {2, 3, 6, 9} is

Choose the correct answer

15. Consider \(a_{33}\) it is the biggest number in the \(3 \times 3\) sub matrix formed by top left corner so minimum value of \(a_{33}\) is 9. Now again consider \(a_{33}\) it is the smallest number in \((n-3)(n-3)\) sub matrix formed by bottom left corner of the main matrix. If \(b_j\) denotes the number of elements in the set \(\{a_{jj} : j^2 \leq a_{jj} \leq n^2 - (n-j+1)^2 + 1\}\), find \(\sum_{j=1}^{n} b_j\).

If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, then $(A + B)^9$ is equal to

Let $A_k=[a_{ij}]$ be square matrix of order 3 with $a_{ij}=(i-j)^k$ for all $i,j\in\{1,2,3\}$. Determinant value of $|A_1+A_3+A_5+\cdots+A_{2023}|$ equals

Solve the system $x+y+z=6$, $x-y+z=2$, $2x+y-z=1$ using matrix method.

Let \(A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}\) and \(A^2 = \begin{bmatrix} \alpha & \beta \\ \beta & \alpha \end{bmatrix}\). Then which of the following is correct?