Matrices & Determinants Questions - JEE

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$ and $P = \left[p_{ij}\right]$ be a $n\times n$ matrix with $p_{ij} = \omega^{i+j}$. Then $P^2 \neq 0$ when $n =$

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 $A = \begin{pmatrix} a & b \\ b & a \end{pmatrix}$ and $A^2 = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix}$, then

Let $X$ and $Y$ be two arbitrary, $3 \times 3$, non-zero skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$, non-zero symmetric matrix. Then which of the following is (are) skew-symmetric:

Which of the following is (are) not the square of a $3 \times 3$ matrix with real entries:

The system of equations $6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0$ has:

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:

The determinant $$\begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix}$$ is equal to zero, if:

If $A = \begin{bmatrix} 1 & 3 \\ 3 & 2 \\ 2 & 5 \end{bmatrix}$ & $B = \begin{bmatrix} -1 & -2 \\ 0 & 5 \\ 3 & 1 \end{bmatrix}$ and $A + B - D = O$ (zero matrix), then $D$ matrix will be-

If $a, b, c$ are non-zero real numbers such that $$\begin{vmatrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \end{vmatrix} = 0$$, then:

A is the n × n matrix whose elements are all '1' and B is the n × n matrix whose diagonal elements are all 'n' and other elements are 'n − r'. Then, A² is a scalar multiple of A and then $(B - rI)[B - (n^2 - nr + r)I]$ is

Let $\{\Delta_1, \Delta_2, \Delta_3, \ldots \Delta_k\}$ be the set of third order determinants that can be made with the distinct non-zero real numbers $a_1, a_2, \ldots a_9$. Then:

Let $B = \dfrac{1}{10}\begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{bmatrix}$ be the inverse of $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$. Then $\alpha$ equals:

For $3 \times 3$ Matrices $M$ and $N$, which of the following statement(s) is (are) not correct

94. Let A and B are square matrices of same order satisfying $AB = A$ and $BA = B$, then $(A^{2019} + B^{2019})^{2020}$ is equal to:

If determinant $\Delta$ is non-zero, then matrix is:

If $A$ is square matrix of order $n$ then $\text{adj}(\text{adj } A) =$

If $\det(A)$=5, then det(A\)^6(I\) - $A$⁻^6)(I\) + $A$⁻^6)(I\) - $A$⁻^6)(I\) + $A$⁻^6)) equals:

If $\det(A)$=5, then det(A\)^6(A\)⁻^6 - \(I)) equals:

If $\phi(x) = \begin{vmatrix} 2 & 3 & -1 \\ \log_e(1+x^2) & e^x & \sin x \\ \cos x & \tan x & \sin x \end{vmatrix}$, for $x\geq 0$, then

For any real values of $X, Y, Z, L, M, N$ value of $\begin{vmatrix} \cos(X - L) & \cos(X - M) & \cos(X - N) \\ \cos(Y - L) & \cos(Y - M) & \cos(Y - N) \\ \cos(Z - L) & \cos(Z - M) & \cos(Z - N) \end{vmatrix} =$

If $\Delta$ = 5 and one row is multiplied by 3, new determinant is:

If $\det(A)$=4, then det(A\)^5(A\)⁻^5 - \(I)) equals:

Let $f(x) = \begin{vmatrix} x \cos x & 2x \sin x & x \tan x \\ 1 & 2x & 1 \end{vmatrix}$, then $\lim_{x \to 0} \frac{f(x)}{x^2} =$

996. Let $P$ be a $2 \times 2$ matrix such that $P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}$ and $P^2\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}$. If $x_1$ and $x_2$ are two values of $x$ for which $|P - xI| = 0$, where $I$ is an identity matrix of order 2, then find the value of $x_1^2 + x_2^2$.

If $\det(A)$=5, then det(A\)^6(A\)⁻^6 + $I)(A$⁻^6 - \(I)) equals:

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:

$a+d=$

If $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$, $P = \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$, $Q = P^T AP$, then $PQ^{2014}P^T =$

For $3 \times 3$ Matrices $M$ and $N$, which of the following statement(s) is (are) not correct

The ratio of the trace of matrix B to matrix C is (where C = (A ⋅ B))

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

If $A$ is a square matrix of order $n \times n$ and $k$ is a scalar, then $adj(kA)$ is equal to

If $A = [a_{ij}]_{4 \times 4}$, such that $a_{ij} = \begin{cases} 2, & \text{when } i = j \\ 0, & \text{when } i \neq j \end{cases}$, then $\left\{ \dfrac{\det(\text{adj}(\text{adj } A))}{7} \right\}$ is (where $\{\cdot\}$ represents fractional part function)

The determinant $$\begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix}$$ is equal to zero, if:

If $A, B$ are two matrices such that $A + B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}, A - B = \begin{bmatrix} 3 & 2 \\ -2 & 0 \end{bmatrix}$, then find $AB$.

For the system $x+y+z=6$, $\alpha x+\beta y+7z=3$, $x+2y+3z=14$, which of the following is NOT true?

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

If $a, b, c$ are non-zero real numbers such that $$\begin{vmatrix} bc & ca & ab \\ ca & ab & bc \\ ab & bc & ca \end{vmatrix} = 0$$, then:

If $ax + by + cz = 0$, $bx + cy + az = 0$, $cx + ay + bz = 0$, $a,b,c \in \mathbb{R}^+$ then:

Consider the system $\alpha x+2y+z=1$, $2\alpha x+3y+z=1$, $3x+\alpha y+2z=\beta$, $\alpha,\beta\in\mathbb{R}$. Which of the following is NOT correct?

The system of equations $6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0$ has:

If $\det(A)$=k, then det(A\)⁻^nA^m) where n>m equals:

Let An and Bn be square matrices of order 3, which are defined as:An = [aij] and Bn = [bij] where $a_{ij} = \frac{2i+j}{3 \cdot 2^n}$ and $b_{ij} = \frac{3i-j}{2^n}$ for all i and j, $1 \leq i, j \leq 3$.If $\lambda = \lim_{n \to \infty} \text{Tr.}(3A_1 + 3^2A_2 + 3^3A_3 + \cdots + 3^nA_n)$ and$\mu = \lim_{n \to \infty} \text{Tr.}(2B_1 + 2^2B_2 + 2^3B_3 + \cdots + 2^nB_n)$, then find the value of $\frac{\lambda + \mu}{3}$[Note: Tr.(P) denotes the trace of matrix P.]

If $\det(A)$=k, then $\det(I)$ equals:

If the system of linear equations$x + 2ay + az = 0$$x + 3by + bz = 0$$x + 4cy + cz = 0$has a non-zero solution, then $a, b, c$

Let the system $x+y+kz=2$, $2x+3y-z=1$, $3x+4y+2z=k$ have infinitely many solutions. Then the system $(k+1)x+(2k-1)y=7$, $(2k+1)x+(k+5)y=10$ has:

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible, if:

If $S = \left\{ x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0 \right\}$, then $\displaystyle\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)$ is equal to

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:

Matrices & Determinants Questions (2045)