Matrices & Determinants Questions - JEE

If $\Delta = \begin{vmatrix} \sin x & \sin(x+h) & \sin(x+2h) \\ \sin(x+2h) & \sin x & \sin(x+h) \\ \sin(x+h) & \sin(x+2h) & \sin x \end{vmatrix}$, find $\lim_{h \to 0} \frac{\Delta}{h^2}$.

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

If f(x) = \begin{vmatrix} \sin x & \cos x & \sin x \\ \cos x & -\sin x & \cos x \\ x & 1 & 1 \end{vmatrix}, find the value of 2 \cdot \frac{f'(0)}{[f'(1)]^2}

Let \[A = \begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}\] Then which of the following is true?

For two $ 3 \times 3 $ matrices $ A $ and $ B $, let $ A + B = 2B' $ and $ 3A + 2B = I_3 $, where $ B' $ is the transpose of $ B $ and $ I_3 $ is $ 3 \times 3 $ identity matrix. Then

Let $A$ be a matrix such that $A\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ is a scalar matrix and $|3A| = 108$. Then $A^2$ equals

Let $xC_i$, $x^2C_i$ and $x^3C_i$ ($i \in \{1, 2, 3\}$) be Binomial coefficients, where $x \in \mathbb{N}$and \[f(x) = \begin{vmatrix} xC_1 & xC_2 & xC_3 \\ x^2C_1 & x^2C_2 & x^2C_3 \\ x^3C_1 & x^3C_2 & x^3C_3 \end{vmatrix}\]Then $f(x)$ is a polynomial of degree:

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The system of equations $x_1 - x_2 + x_3 = 2, 3x_1 - x_2 + 2x_3 = -6$ and $3x_1 + x_2 + x_3 = -18$ has

If $\alpha$, $\beta$, $\gamma$ are constants, then the value of\[\sum_{r=1}^{n} S_r = \begin{vmatrix} \sum_{r=1}^{n} 2^{r-1} & \alpha & 2^{n-1} \\ \sum_{r=1}^{n} 2 \cdot 3^{r-1} & \beta & 3^{n-1} \\ \sum_{r=1}^{n} 4 \cdot 5^{r-1} & \gamma & 5^{n-1} \end{vmatrix}\]equals

If $A_i = \begin{pmatrix} 2 - i & 3 - i \\ 3 & 2 \end{pmatrix}$, then $\sum_{i=1}^{\infty} \det(A_i)$ is equal to

If the value of the determinant $\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}$ is positive, then $(a, b, c > 0)$

If A = $$\begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix}$$, then A is

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and each element is multiplied by k, determinant becomes:

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, $Z = PQ^{-1}$

Matrices & Determinants Questions (2045)