Determinants Questions (2072)

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 \(Q\).

For a unique value of m and l, the system of equations given by\(x + y + z = 6\)\(x + 2y + 3z = 14\)\(2x + 5y + lz = m\)has infinitely many solutions, then \(\frac{m - l}{4}\) is equal to

Let \(A = \begin{bmatrix}i & 0\\0 & i\end{bmatrix}\). Find the trace of \(A^{4n}\).

Point P(x, y) is rotated by an angle θ in anticlockwise direction. The new position of point P is Q(x1, y1). If \(\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = A \begin{bmatrix} x \\ y \end{bmatrix}\), then find matrix A.

If \(AB = \dfrac{6}{8}\) and \(\begin{bmatrix}1 & 2 & x \\ 3 & -1 & 2\end{bmatrix}_{2\times3} \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}_{3\times1} = \begin{bmatrix}6\\8\end{bmatrix}\), find \(y\).

If A is symmetric as well as skew-symmetric matrix, then A is

If the system of linear equations\((\cos\theta) x + (\sin\theta) y + \cos\theta = 0\)\((\sin\theta) x + (\cos\theta) y + \sin\theta = 0\)\((\cos\theta) x + (\sin\theta) y - \cos\theta = 0\)is consistent, then the number of possible values of \(\theta\), \(\theta \in [0, 2\pi]\) is:

If \(a\), \(b\) and \(c\) are the roots of the equation \(x^3 + 2x^2 + 1 = 0\), find \begin{vmatrix} a & b & x \\ b & c & a \\ c & a & b \end{vmatrix}.

Let \(A\) be a square matrix all of whose entries are integers. Then which one of the following is true?

If \(A = \begin{bmatrix}1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b\end{bmatrix}\) and \(A \cdot A^T = 9I\), then find the values of \(a\) and \(b\).

Consider the system of linear equations:\(x_1 + 2x_2 + x_3 = 3\)\(2x_1 + 3x_2 + x_3 = 3\)\(3x_1 + 5x_2 + 2x_3 = 1\)The system has

If \(A\), \(B\), \(C\) are the angles of triangle \(ABC\), then the minimum value of \begin{vmatrix} -2 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix} is equal to:

If \(a = \cos\theta + i\sin\theta\), \(b = \cos 2\theta - i\sin 2\theta\), \(c = \cos 3\theta\) \(+ i\sin 3\theta\) and if \(\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0\), then

Consider the system of linear equations $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4\lambda^2 z=\mu^2+15$, where $\lambda,\mu\in\mathbb{R}$. Which one of the following statements is NOT correct?

Let $d$ be a matrix of order $3 \times 3$ such that $\det(4) = 2$, $B = 2d$ and $C = \frac{1}{\sqrt[3]{4}}$, then the value of $\det\left(\frac{d^3(C^3)}{\sqrt[3]{d}}\right)$ is