Determinants Questions - JEE

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

Given $f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$, find $f'(x)$.

Three planes are given by:x + 4y - 2z = 1x + 7y - 5z = bx + 5y + az = 5If their intersection is a line in \mathbb{R}^3, find the values of a and b.

If $f(n) = \alpha^n + \beta^n$ 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 $k\in\mathbb{Z}$ is equal to

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

If $\mathbf{A} = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, then the matrix A is

If AT + A = I, where A = $\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$, find the value of $\theta$.

Given, $a^n b^{2n-2}$ where $\begin{vmatrix} 1 & a^2 & a^4 \\ 1 & b^2 & b^4 \\ 1 & c^2 & c^4 \end{vmatrix}$ and on comparison, we get $n = -2$

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

Let $A = \begin{bmatrix} 1 & -3 & 2 \\ 2 & 1 & -3 \\ 4 & -3 & -1 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 4 & 1 & 0 \\ 2 & 1 & 1 & 1 \\ 1 & -2 & 1 & 2 \end{bmatrix}$ & $C = \begin{bmatrix} 1 & 1 & -1 & -2 \\ 3 & -2 & -1 & -1 \\ 2 & -5 & -1 & 0 \end{bmatrix}$ be the matrices then, prove that in matrix multiplication cancellation law does not hold.

Which of the following holds true for matrix multiplication?

Given set, A = { X = (x, y, z)T : PX = 0 and x2 + y2 + z2 = 1, where P = \[\begin{pmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{pmatrix}\] }. Find the number of elements in set A.

For Problems 4–6If $A$ and $B$ are two square matrices of order $3 \times 3$ which satisfy $AB = A$ and $BA = B$, thenWhich of the following is true?

Let $S$ be the set of all values of $\theta\in[-\pi,\pi]$ for which the system $x+y+\sqrt{3}z=0$, $-x+(\tan\theta)y+\sqrt{7}z=0$, $x+y+(\tan\theta)z=0$ has non-trivial solution. Then $\dfrac{120}{\pi}\displaystyle\sum_{\theta\in S}\theta$ is equal to

If det($A$) = 2, then det($A^3$) is:

Elements of a matrix A of order 10 × 10 are defined as aij = ωi+j (where ω is cube root of unity), then trace(A) of the matrix is

Numerical question placeholder

Let $A$ be a square matrix of order $3$ such that $\det(A)=-2$ and $\det\bigl(3\operatorname{adj}(-6\operatorname{adj}(3A))\bigr)=2^{m+n}\cdot 3^{mn},\ m>n$. Then $4m+2n$ equals \rule{2cm}{0.4pt}.

The sum of values of p for which the equations $x + y + z = 1$, $x + 2y + 4z = p$, and $x + 4y + 10z = p^2$ have a solution is ________.

Given $A = \sum_{i=1}^{4} A_i$ where $A_i = \sum_{r=i}^{4} r \cdot C_r$, $\sum_{n=2}^{2} = $ and $\sum_{n=2}^{4} (4 - r) \cdot C_r$. Now, $\sum_{r=1}^{4} r \cdot C_r = 4 \times 2^3 = 32$. Find $|A| = 1024 - 150 = 874$.

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

760. Let $A = P^{-1}DP$ where $D = \text{diag}(1,2,3)$. Find $\det(A^2 + A)$.

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

Let $A=\begin{pmatrix}1&5/1\\0&1\end{pmatrix}$, $B=\begin{pmatrix}1&2\\-1&-1\end{pmatrix}A\begin{pmatrix}-1&-2\\1&1\end{pmatrix}$, then the sum of all entries of $\displaystyle\sum_{n=1}^{50}B^n$ is equal to

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$, where $b > 0$. Then, the minimum value of $\frac{\det(A)}{b}$ is

If $\begin{vmatrix}x+1&x&x\\x&x+\lambda&x\\x&x&x+\lambda^2\end{vmatrix}=\dfrac{9}{8}(103x+81)$, then $\dfrac{\lambda}{8}$ and $\dfrac{\lambda}{3}$ are the roots of the equation

If $A = \begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}$, $n \in \mathbb{N}$, then trace of $A^{4n}$ equals to ___________.

Let $A$ be a $2\times2$ matrix with real entries such that $A^\top=\alpha A+I$, where $\alpha\in\mathbb{R}\setminus\{-1,1\}$. If $\det(A^2-A)=4$, the sum of all possible values of $\alpha$ is equal to

Let A and B be matrices of order $3 \times m$ and $n \times 3$ respectively. If $A + B$ as well as $AB$ exist then

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

Let $M$ denote the set of all real matrices of order $3\times 3$ and let $S=\{-3,-2,-1,1,2\}$. $S_{1}=\{A\in M\,:\,A=A^{T},\ a_{ij}\in S\}$, $S_{2}=\{A\in M\,:\,A=-A^{T},\ a_{ij}\in S\}$, $S_{3}=\{A=[a_{ij}]\in M\,:\,a_{11}+a_{22}+a_{33}=0,\ a_{ij}\in S\}$. If $n(S_{1}\cup S_{2}\cup S_{3})=125\alpha$, then $\alpha$ equals \rule{2cm}{0.4pt}.

Let $A=\begin{pmatrix}2&1&0\\1&2&-1\\0&-1&2\end{pmatrix}$. If $|\text{adj}(\text{adj}(\text{adj}\,2A))|=(16)^n$, then $n$ is equal to

Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ²z = μ² + 15 where λ, μ ∈ R. Which one of the following statements is NOT correct ? [JEE (Main) 2024]

If three distinct points $P(3u^2, 2u^3)$, $Q(3v^2, 2v^3)$, and $R(3w^2, 2w^3)$ are collinear, then $uv + vw + wu$ is equal to ________.

Let 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 $\text{tr}(A) + \text{tr}\!\left(\dfrac{A(BC)}{2}\right) + \text{tr}\!\left(\dfrac{A(BC)^2}{4}\right) + \text{tr}\!\left(\dfrac{A(BC)^3}{8}\right) + \cdots\infty$ is equal to ___________.

Given, $Z = PQ^{-1}$

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:

Determinants Questions (2072)