Matrices & Determinants Questions - JEE

If w ≠ 1 is the complex cube root of unity and matrix H = ⎡⎣1 0⎤⎦0 w, then H70 is equal to

If \[\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\] where $i = \sqrt{-1}$, then

If $A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}$ and B is the adjoint of A, find the value of $|AB + 2I|$, where I is the identity matrix of order 3.

If a, b, c, d, e, and f are in G.P., then the value of $\begin{vmatrix} a^2 & d^2 & x \\ b^2 & e^2 & y \\ c^2 & f^2 & z \end{vmatrix}$ depends on

Let $A = \{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\}$, where \[P = \begin{pmatrix}1 & 2 & 1\\-2 & 3 & -4\\1 & 9 & -1\end{pmatrix}\]Then the set A

The system of linear equations $x+y+z=6$ $2x+5y+az=36$ $x+2y+3z=b$ has

The number of values of $k$ for which the linear equations$4x + ky + 2z = 0$$kx + 4y + z = 0$$2x + 2y + z = 0$possess a non-zero solution is

If A is a square matrix such that A2 = A, then det(A) is equal to

If the system of linear equations$2x + 2y + 3z = a$$3x - y + 5z = b$$x - 3y + 2z = c$where $a, b, c$ are non-zero real numbers, has more than one solution, then:

If $u + 2v + 3w = 6$, $4u + 5v + 6w = 12$, and $6u + 9v = 4$, then $u + v + w$ is equal to

If $1 + \sin 2x$ $\cos 2x$ $4\sin^2 x$ $\sin^2 x$ $1 + \cos^2 x$ $4\sin^4 x$ $\sin x$ $\cos x$ $1 + 4\sin^2 x$ is evaluated, the maximum value is

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 $\dfrac{\det(A)}{b}$ is:

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix} \] and x = a + 2p, y = b + 2q, z = c + 2r, then $\Delta$ equals:

The total number of distinct $x \in \mathbb{R}$ for which\[\begin{vmatrix} x & x^2 & 1+x^3 \\ 2x & 4x^2 & 1+8x^3 \\ 3x & 9x^2 & 1+27x^3 \end{vmatrix} = 10\]is ______.

The value of the determinant $\begin{vmatrix} 1 & 1 & 1 \\ ^mC_1 & ^{m+1}C_1 & ^{m+2}C_1 \\ ^mC_2 & ^{m+1}C_2 & ^{m+2}C_2 \end{vmatrix}$ is equal to

The sum of number of elements of $m_{n-1}$ matrices gives the last element of $m_{n-1}$ matrix. The last element of $m_{n-1}$ matrix is $1^2+2^2+3^2+\cdots+(n-1)^2 = \dfrac{n(n-1)(2n-1)}{6}$. The first element of $m_{10}$ matrix is $\dfrac{10 \times 9 \times 19}{6}+1 = 286$. The common difference in diagonal elements of $m_n$ matrix is $n+1$; in $m_{10}$, the common difference is 11. Find the sum of diagonal elements of $m_{10}$.

For a real number a, if the system\[\begin{pmatrix} 1 & a & a \\ a & 1 & a \\ a & a & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}\]of linear equations has infinitely many solutions, then $1 + a + a^2 = $ ______.

If the system of linear equations \[x + ky + 3z = 0\]\[3x + ky - 2z = 0\]\[2x + 4y - 3z = 0\] has a non-zero solution $(x, y, z)$, then $\frac{xz}{y^2}$ is equal to

If $f(x) = \begin{vmatrix} 1 & x & x^2 & x^3 \\ 0 & 1 & 2 & 3 \\ 0 & 1 & x & x \end{vmatrix}$, then $8f(1)$ is equal to

If A, B and A + B are idempotent matrices, then AB is equal to

If A is a diagonal matrix of order 3 × 3 is commutative with every square matrix of order 3 × 3 under multiplication and trace(A) = 12, then

If $p\sqrt{2} + 3\sqrt{4} + q\sqrt{3} + r\sqrt{2} + s\sqrt{1} + t = \sqrt{2} + 1$ and $\begin{vmatrix} 2+3\sqrt{4} & \sqrt{4}-1 & \sqrt{4}+3 \\ \sqrt{2}+1 & 2-\sqrt{4} & \sqrt{4}-3 \\ \sqrt{2}-3 & \sqrt{4}+4 & 3\sqrt{4} \end{vmatrix}$, then $t$ is equal to

The determinant abaa+bbcba+caa+bba+c0 is equal to zero, if -

The system of equations x + y + z = 2, 2x + y - z = 3, 3x + 2y + \lambda z = 4 has unique solution if

If $|\text{adj}(\text{adj}\, A)| = |A|^{n-2}$, then for a square matrix $A$ of order $n$, which of the following is true?We have $|\text{adj}(A)| = |A|^{n-1}$ and $\text{adj}(\text{adj}\, A) = |A|^{n-2} A$.

Let $px^4 + qx^3 + rx^2 + sx + t = \begin{vmatrix} x+1 & -2x & x-4 \\ x^2+3x & x-1 & x+3 \\ x-3 & x+4 & 3x \end{vmatrix}$, where $p, q, r, s$ and $t$ are constants, then $t$ is equal to

If $f(x), g(x)$ and $h(x)$ are polynomials of degree 4 and $\begin{vmatrix} f(x) & g(x) & h(x) \\ a & b & c \\ p & q & r \end{vmatrix} = mx^4 + nx^3 + rx^2 + sx + t$ be an identity in $x$, then $\frac{f'''(0) - f''(0)}{a} + \frac{g'''(0) - g''(0)}{b} + \frac{h'''(0) - h''(0)}{c}$ is equal to

If A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}\end{pmatrix} and \det(A^n - I) = 1 - \lambda^n, n \in \mathbb{N}, then the value of \lambda is

The matrix \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} is the matrix reflection in the line

If $A = \begin{bmatrix}-1 & x \\ 4 & 2\end{bmatrix}$, $B = \begin{bmatrix}x & 1 \\ -1 & 0\end{bmatrix}$ and $AB = \begin{bmatrix}-6 & -1 \\ 10 & 4\end{bmatrix}$ then the value of $x$ is

Let $\theta = \frac{\pi}{5}$ and $A = \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix}$. If $B = A + A^4$, then $\det(B)$ is

Let $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. If $B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$, then the sum of all the elements of the matrix $\sum_{n=1}^{50} B^n$ is equal to

Let $A$ be a $3\times3$ matrix such that $A+A^T=O$. If $A\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}3\\3\\2\end{bmatrix}$, $A^2\begin{bmatrix}1\\-1\\0\end{bmatrix}=\begin{bmatrix}-3\\19\\-24\end{bmatrix}$ and $\det(\text{adj}(2\,\text{adj}(A+I)))=(2)^\alpha\cdot(3)^\beta\cdot(11)^\gamma$, $\alpha,\beta,\gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to _____.

The values of x for which the given matrix $\begin{pmatrix} -x & x & 2 \\ 2 & x & -x \\ x & -2 & -x \end{pmatrix}$ will be non-singular are

If the matrices A = \begin{pmatrix} 1 & 1 & 2 \\ 1 & 3 & 4 \\ 1 & -1 & 3 \end{pmatrix}, B = \text{adj } A and C = 3A, then |\text{adj } B| is equal to

The value of a for which the following system of equations a^3 x + (a+1)^3 y + (a+2)^3 z = 0, ax + (a+1)y + (a+2)z = 0, x + y + z = 0 has a non-trivial solution is equal to

Given that matrix B is the inverse of a $3 \times 3$ matrix A, where:\[B = \begin{pmatrix} 5 & 2a & 1 \\ 0 & 2 & 1 \\ a & 3 & -1 \end{pmatrix}\]If $\det(A) + 1 = 0$, find the value(s) of $a$.

Let k be a positive real number and let\[A = \begin{pmatrix} 2k-1 & 2k & 2k \\ 2k & 1 & -2k \\ -2k & 2k & -1 \end{pmatrix}\]and\[B = \begin{pmatrix} 0 & 2k-1 & k \\ 1-2k & 0 & 2k \\ -k & -2k & 0 \end{pmatrix}\]If $\det(\text{adj } A) + \det(\text{adj } B) = 106$, then $[k]$ is equal to ______.

If f(θ) = |1tanθ1−tanθ1tanθ−1−tanθ1|, then the set {θ:f(θ)=0,θ∈[−π,π]} is

Let A and B be two non-singular matrices such that $A \neq I$, $B^3 = I$ and $AB = BA^2$, where I is the identity matrix. Find the least value of k such that $A^k = I$.

Let A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, if u_1 and u_2 are column matrices such that Au_1 = 0 and Au_2 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, then u_1 + u_2 is

If $\sin^2\theta (1 + \sin^2\theta)$ $\cos^2\theta$ $4\sin^4\theta$ $\sin^2\theta$ $1 + \cos^2\theta$ $4\sin^4\theta$ $\sin\theta$ $\cos\theta$ $1 + 4\sin^4\theta$ = 0, then $\theta$ 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 the system of linear equations $x-2y+z=-4$, $2x+\alpha y+3z=5$, $3x-y+\beta z=3$ has infinitely many solutions, then $12\alpha+13\beta$ is equal to

The set of all values of $\lambda$ for which the system of linear equations$2x_1 - 2x_2 + x_3 = \lambda x_1$$2x_1 - 3x_2 + 2x_3 = \lambda x_2$$-x_1 + 2x_2 = \lambda x_3$has a non-trivial solution,

Let $A$ be a $3\times3$ real matrix such that $A\begin{pmatrix}1\\0\\1\end{pmatrix}=2\begin{pmatrix}1\\0\\1\end{pmatrix}$, $A\begin{pmatrix}-1\\0\\1\end{pmatrix}=4\begin{pmatrix}-1\\0\\1\end{pmatrix}$, $A\begin{pmatrix}0\\1\\0\end{pmatrix}=2\begin{pmatrix}0\\1\\0\end{pmatrix}$. Then, the system $(A-3I)\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}$ has

If the system of equations $2x+3y-z=5$ $x+\alpha y+3z=-4$ $3x-y+\beta z=7$ has infinitely many solutions, then $13\alpha\beta$ is equal to

For a real number $\alpha$, if the system \[\begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\] of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 =$ ___. (JEE Advanced 2017)

The set of natural numbers is divided into arrays of rows and columns in the form of matrices as: $A_1 = [1]$, $A_2 = \begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$, $A_3 = \begin{pmatrix} 6 & 7 & 8 \\ 9 & 10 & 11 \\ 12 & 13 & 14 \end{pmatrix}$ and so on. Let the trace of A10 be λ. Find the unit digit of λ.

Given $P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $Q - P^5 = I_3$. Find $Q$.

Matrices & Determinants Questions (2045)