Matrices & Determinants Questions (2045)

If the matrix \(\begin{pmatrix}0.3 & b & c \\ l & m & n \\ 0 & p & q\end{pmatrix}\) is an orthogonal matrix, find the sum of all possible values of \(10(mq - np)\).

Let $A$ be a square matrix such that $AA^T=I$. Then $\dfrac{1}{2}A\left[(A+A^T)^2+(A-A^T)^2\right]$ is equal to

If \(x = 31\), \(y = 32\) and \(z = 33\), then the value of \[\begin{vmatrix} (x^2+1)^2 & (xy+1)^2 & (xz+1)^2 \\ (xy+1)^2 & (y^2+1)^2 & (yz+1)^2 \\ (xz+1)^2 & (yz+1)^2 & (z^2+1)^2 \end{vmatrix}\] is ________.

If \(a, b\) and \(c\) are sides of a triangle and \(\begin{vmatrix} a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2 \end{vmatrix} = 0\), then

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

Let the system $-x+2y-9z=7$, $-x+3y+7z=9$, $-2x+y+5z=8$, $-3x+y+13z=\lambda$ have a unique solution $x=\alpha,\ y=\beta,\ z=\gamma$. Then the distance of the point $(\alpha,\beta,\gamma)$ from the plane $2x-2y+z=\lambda$ is

The system of equations x + y + z = 5; x + 2y + 3z = 9; x + 3y + Dz = I is called good, if it has infinitely many solutions. The condition for this is

If the system of equations $x+(\sqrt{2}\sin\alpha)y+(\sqrt{2}\cos\alpha)z=0$, $x+(\cos\alpha)y+(\sin\alpha)z=0$, $x+(\sin\alpha)y-(\cos\alpha)z=0$ has a non-trivial solution, then $\alpha\in\left(0,\dfrac{\pi}{2}\right)$ is equal to:

Let $S$ be the set of values of $\lambda$ for which the system $6\lambda x-3y+3z=4\lambda^2$, $2x+6\lambda y+4z=1$, $3x+2y+3\lambda z=\lambda$ has no solution. Then $12\displaystyle\sum_{\lambda\in S}|\lambda|$ is equal to _______.

Using properties of determinants, evaluate\[\begin{vmatrix} 18 & 40 & 89 \\ 40 & 89 & 198 \\ 89 & 198 & 440 \end{vmatrix}\]

If the system $2x+y-z=5$, $2x-5y+\lambda z=\mu$, $x+2y-5z=7$ has infinitely many solutions, then $(\lambda+\mu)^2+(\lambda-\mu)^2$ is equal to

Let $P$ be a square matrix such that $P^2=I-P$. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $P^\alpha+P^\beta=\gamma I-29P$ and $P^\alpha-P^\beta=\delta I-13P$, then $\alpha+\beta+\gamma-\delta$ is equal to

If $adj(adjM) = |M|^{n-2}M$, find the order of $M$.

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

The values of $m,n$ for which the system of equations $x+y+z=4$, $2x+5y+5z=17$, $x+2y+mz=n$ has infinitely many solutions, satisfy the equation:

If $\alpha\neq a$, $\beta\neq b$, $\gamma\neq c$ and $\begin{vmatrix}\alpha&b&c\\a&\beta&c\\a&b&\gamma\end{vmatrix}=0$, then $\dfrac{a}{\alpha-a}+\dfrac{b}{\beta-b}+\dfrac{\gamma}{\gamma-c}$ is equal to:

69. If \(S_r = \begin{vmatrix} 2^{r-1} & \alpha & 2^{n-1} \\ 2 \cdot 3^{r-1} & \beta & 3^{n-1} \\ 4 \cdot 5^{r-1} & \gamma & 5^{n-1} \end{vmatrix}\), then \(\displaystyle\sum_{r=1}^{n} S_r =\) ______.

Let $A=\begin{bmatrix}2&1&2\\6&2&11\\3&3&2\end{bmatrix}$ and $P=\begin{bmatrix}1&2&0\\5&0&2\\7&1&5\end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP-2I|$ is equal to

Let $M$ and $N$ be square matrices of the same order satisfying $MN = M$ and $NM = N$. Then $(M^{2024} + N^{2024})^{2025}$ is equal to

Consider three matrices \(A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}\), \(B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}\), and \(C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}\). Then the value of the sum \(\text{tr}(A) + \text{tr}\left(\dfrac{ABC}{2}\right) + \text{tr}\left(\dfrac{A(BC)^2}{4}\right) + \text{tr}\left(\dfrac{A(BC)^3}{8}\right) + \cdots + \infty\) is

Let $S=\{m\in\mathbb{Z}\,:\,A^{m^{2}}+A^{m}=3I-A^{-6}\}$, where $A=\begin{pmatrix}2&-1\\1&0\end{pmatrix}$. Then $n(S)$ is equal to:

If the system of equations $11x+y+\lambda z=-5$, $2x+3y+5z=3$, $8x-19y-39z=\mu$ has infinitely many solutions, then $\lambda^4-\mu$ is equal to:

Given \(A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}\) and \(A\,\text{adj}\,A = AA^T\), find \(5a + b\).

Let $\alpha\beta\gamma=45$; $\alpha,\beta,\gamma\in\mathbb{R}$. If $x(\alpha,1,2)+y(1,\beta,2)+z(2,3,\gamma)=(0,0,0)$ for some $x,y,z\in\mathbb{R}$, $xyz\neq0$, then $6\alpha+4\beta+\gamma$ is equal to ________.

258. If \(A = \begin{bmatrix} a & x & y \\ x & b & z \\ y & z & c \end{bmatrix}\) where \(a, b, c, x, y, z \in \{1, 2, 3, 4, 5, 6\}\) and also \(a, b, c, x, y, z\) are distinct, then number of matrices in \(A\) with trace equal to 10 are:

If \(AB = O\) for the matrices \(A = \begin{bmatrix} \cos^2\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta \end{bmatrix}\) and \(B = \begin{bmatrix} \cos^2\phi & \cos\phi\sin\phi \\ \cos\phi\sin\phi & \sin^2\phi \end{bmatrix}\), then \(\theta - \phi\) is ___________ (in degree).

If $A$ is $3\times3$ and $|A|=2$, then $\left|3\,\text{adj}(|3A|A^2)\right|$ is equal to

If \(\det(A)\)=2, then det(A\)^2A^3A⁻^4A^{-1}) equals:

Let \(A\) be an \(m \times m\) matrix with all elements equal to 1 such that \(A^n = 16^{17}\, A\), \(m, n \in N\). Find the sum of possible values of \(n\).

$A = adj(B = adj(adj 4))$ where $|A|\cdot|A| = |A|\cdot|B| = 1$

Let $A=[a_{ij}]$ be a matrix of order $3\times 3$ with $a_{ij}=(\sqrt{2})^{i+j}$. If the sum of all the elements in the third row of $A^{2}$ is $\alpha+\beta\sqrt{2},\,\alpha,\beta\in\mathbb{Z}$, then $\alpha+\beta$ is equal to:

If \(\alpha, \beta \neq 0\), and \(f(n) = \alpha^n + \beta^n\) and \[\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2,\] then \(K\) is equal to

If \(\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix} = (a+b+c)(x+a+b+c)^2\), where \(x \neq 0\) and \(a+b+c \neq 0\), then \(x\) is equal to

The system of equations $x + y + z = 6$, $x + 2y + 5z = 9$, $x + 5y + \lambda z = \mu$ has no solution if:

If \(\det(A)\)=4, then det((A\)^2)^{-1}) equals:

If \(\det(A)\)=2, then det(A\)⁻^3A^5A^{-1}\(A\)^2) equals:

If the system $(\lambda-1)x+(\lambda-4)y+\lambda z=5,\ \lambda x+(\lambda-1)y+(\lambda-4)z=7,\ (\lambda+1)x+(\lambda+2)y-(\lambda+2)z=9$ has infinitely many solutions, then $\lambda^{2}+\lambda$ is equal to:

Let $A$ be a $3 \times 3$ matrix such that $X^TAX = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix}x\\y\\z\end{bmatrix}$. If $A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}$, $A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}$, and $\det(\text{adj}(2A + I)) = 2^\alpha 3^\beta 5^\gamma$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is ___

Let $\alpha,\beta\,(\alpha\ne\beta)$ be the values of $m$ for which the equations $x+y+z=1,\ x+2y+4z=m,\ x+4y+10z=m^{2}$ have infinitely many solutions. Then $\displaystyle\sum_{n=1}^{10}(n^{\alpha}+n^{\beta})$ is equal to:

If the system of linear equations $x + y + 2z = 6$, $2x + 3y + az = a + 1$, $-x - 3y + bz = 2b$ where $a, b \in \mathbb{R}$, has infinitely many solutions, then $7a + 3b$ is equal to:

If $A$ is a square matrix of order 3 such that $\det(A)=3$ and $\det(\text{adj}(-4\,\text{adj}(-3\,\text{adj}(3\,\text{adj}((2A)^{-1})))))=2^m3^n$, then $m+2n$ is equal to:

Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x) = \begin{vmatrix}1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x\end{vmatrix}$, $x \in \mathbb{R}$. Then $M^4 - m^4$ is equal to:

Let $A=\begin{pmatrix}-\dfrac{1}{\sqrt{2}} & 1\\ 0 & 1\end{pmatrix}$ and $P=\begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{pmatrix},\,\theta>0.$ If $B=PAP^{T},\,C=P^{T}B^{10}P$ and the sum of the diagonal elements of $C$ is $\dfrac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is:

Let a - 2b + c = 1. If f(x) = x+ax+2x+1x+bx+3x+2x+cx+4x+3, then :

If the system of linear equations $x+y+2z=6,\ 2x+3y+az=a+1,\ -x-3y+bz=2b$, where $a,b\in\mathbb{R}$, has infinitely many solutions, then $7a+3b$ is equal to:

If the system of equations $x + 2y - 3z = 2$, $2x + \lambda y + 5z = 5$, $14x + 3y + \mu z = 33$ has infinitely many solutions, then $\lambda + \mu$ is equal to:

Let $A = [a_{ij}]$ be a $3 \times 3$ matrix such that $A\begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}0\\0\\1\end{bmatrix}$, $A\begin{bmatrix}4\\1\\3\end{bmatrix} = \begin{bmatrix}0\\1\\0\end{bmatrix}$, and $A\begin{bmatrix}2\\1\\2\end{bmatrix} = \begin{bmatrix}1\\0\\0\end{bmatrix}$. Then $a_{23}$ equals:

The values of $\alpha$, for which $\begin{vmatrix}1&\frac{3}{2}&\alpha+\frac{3}{2}\\1&\frac{1}{3}&\alpha+\frac{1}{3}\\2\alpha+3&3\alpha+1&0\end{vmatrix}=0$, lie in the interval

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 $\text{Tr}(A^3)$.

If $A$, $B$, and $\left(\text{adj}(A^{-1}) + \text{adj}(B^{-1})\right)$ are non-singular matrices of same order, then the inverse of $A\left(\text{adj}(A^{-1}) + \text{adj}(B^{-1})\right)^{-1} B$ is equal to: