Determinants Questions (2072)

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:

For some $a$, $b$, let $f(x) = \begin{vmatrix}a + \frac{\sin z}{z} & 1 & b \\ a & 1 + \frac{\sin z}{z} & b \\ a & 1 & b + \frac{\sin z}{z}\end{vmatrix}$, $x \neq 0$, $\lim_{z \to 0} f(x) = \lambda + \mu a + \nu b$. Then $(\lambda + \mu + \nu)^2$ is equal to:

If the system $2x-y+z=4,\ 5x+\lambda y+3z=12,\ 100x-47y+\mu z=212$ has infinitely many solutions, then $\mu-2\lambda$ is equal to:

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

Let $A$ be a $2\times2$ real matrix and $I$ be the identity matrix of order 2. If the roots of the equation $|A-xI|=0$ be $-1$ and $3$, then the sum of the diagonal elements of the matrix $A^2$ is

Let for any three distinct consecutive terms $a,b,c$ of an A.P., the lines $ax+by+c=0$ be concurrent at the point $P$ and $Q(\alpha,\beta)$ be a point such that the system of equations $x+y+z=6$, $2x+5y+\alpha z=\beta$ and $x+2y+3z=4$, has infinitely many solutions. Then $(PQ)^2$ is equal to

If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to:

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

If $A=\begin{bmatrix}\sqrt{2}&1\\-1&\sqrt{2}\end{bmatrix}$, $B=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, $C=ABA^T$ and $X=A^TC^2A$, then $\det X$ is equal to:

Find the condition on $p, q, r$ such that the system of equations:$x + 2y - 3z = p$$2x + 6y - 11z = q$$x - 2y + 7z = r$has infinite solutions.

Let $A = [a_{ij}] = \begin{bmatrix}\log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25\end{bmatrix}$. If $A_{ij}$ is the cofactor of $a_{ij}$, $C_{ij} = \sum_{k=1}^{2} a_{ik} A_{jk}$, $1 \leq i, j \leq 2$, and $C = [C_{ij}]$, then $8|C|$ is equal to:

When the determinant \(\begin{vmatrix} \cos 2x & \sin^2 x & \cos 4x \\ \sin 2x & \cos 2x & \cos 2x \\ \cos 4x & \cos 2x & \cos 2x \end{vmatrix}\) is expanded in powers of \(\sin x\), the constant term in that expression is

For some $a,b$, let $f(x)=\begin{vmatrix}a+\dfrac{\sin x}{x} & 1 & b\\ a & 1+\dfrac{\sin x}{x} & b\\ a & 1 & b+\dfrac{\sin x}{x}\end{vmatrix},\,x\ne 0.$ If $\displaystyle\lim_{x\to 0}f(x)=\lambda+\mu a+\nu b$, then $(\lambda+\mu+\nu)^{2}$ equals:

Let $A=\begin{bmatrix}2&0&1\\1&1&0\\1&0&1\end{bmatrix}$, $B=[B_1,B_2,B_3]$, where $B_1,B_2,B_3$ are column matrices, and $AB_1=\begin{bmatrix}1\\0\\0\end{bmatrix}$, $AB_2=\begin{bmatrix}2\\3\\0\end{bmatrix}$, $AB_3=\begin{bmatrix}3\\2\\1\end{bmatrix}$. If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3+\beta^3$ is equal to

For a $3 \times 3$ matrix $M$, let trace$(M)$ denote the sum of all diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace$(A) = 3$. If $B = \text{adj}(\text{adj}(2A))$, then the value of $|B| + $ trace$(B)$ equals: