Determinants Questions (2072)

The number of diagonal matrices A of order \(n\) for which \(A^3 = A\) is

If \(\Delta\) = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and a, d, g are in A.P., then determinant is:

If \(\begin{vmatrix} a & b-c & c+b \\ a+c & b & c-a \\ a-b & a+b & c \end{vmatrix} = 0\), then the line \(ax + by + c = 0\) passes through the fixed point which is

Let $a, z, y, z$ be real numbers satisfying the equations $az + ay = 5$, $x - ay = z$, $x + ay = az$, where $x, y, z$ are not all zero, then the number of the possible values of $a$ is

If a3 + b3 + c3 − 3abc = −3 and A = bc − a2, B = ca − b2 and C = ab − c2, then the value of aA + bB + cC is

If \(f(x) = \begin{vmatrix} \cos(x-\phi) & \cos(x-\psi) & \cos(x-\omega) \\ \sin(x-\phi) & \sin(x-\psi) & \sin(x-\omega) \\ \sin(\psi - \omega) & \sin(\omega - \phi) & \sin(\phi - \psi) \end{vmatrix}\), then \(f(9) - 2f(6) + f(3)\) is equal to

Find the value of k such that the determinant \[\begin{vmatrix} 3^2 + k & 4^2 & 3^2 + 3 + k \\ 4^2 + k & 5^2 & 4^2 + 4 + k \\ 5^2 + k & 6^2 & 5^2 + 5 + k \end{vmatrix} = 0\]

Question 85: Statement-1: If $f(x) = \begin{vmatrix} (1-x)^{11} & (1-x)^{12} & (1-x)^{13} \\ (1-x)^{21} & (1-x)^{22} & (1-x)^{23} \\ (1-x)^{31} & (1-x)^{32} & (1-x)^{33} \end{vmatrix}$, then the coefficient of $x$ in $f(x) = 0$.Statement-2: If $P(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n$, then $a_1 = P'(0)$, where dash denotes the differential coefficient.

Given matrix $A = \begin{bmatrix} 1 & x & 1 \\ x & 2 & y \\ 1 & y & 3 \end{bmatrix}; B = \begin{bmatrix} 3 & -3 & z \\ -3 & 2 & -3 \\ z & -3 & 1 \end{bmatrix}$. Obtain $x, y, z$ if the matrix $AB$ is symmetric.

Let \(f(x) = \begin{vmatrix} \sec x & x^2 & x \\ 2\sin x & x^3 & 4 \\ \tan 3x & x & x \end{vmatrix}\), then \(\lim_{x \to 0} \frac{f(x)}{x^2}\) is equal to

Sum of real roots of the equation \[\begin{vmatrix} 1 & 4 & 20 \\ 1 & 2 & 5 \\ 1 & 2x & 5x^2 \end{vmatrix} = 0\] is

If \(f(x) = (x-1)^m x^n (x+1)^p\), where \(m, n, p \in \mathbb{N}\), then the value of \(m+n+p\) is:

If the system of equations ax + y = 1, x + 2y = 3, 2x + 3y = 5 are consistent, then a is given by

If A is a square matrix of order 2 such that A−1 = pmatrix1 & -1 \\ -1 & 2pmatrix and Apmatrix2 & 1 \\ 1 & -1pmatrix = pmatrix0pmatrix. The sum of elements and product of elements of A are S and P, then S + P is

If \det(A) = 3, then \det(k\(A\)^{-1}) for \(3 \times 3\) matrix is:

Evaluate the determinant:\(\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix}\)

If I is a unit matrix of order 10, the determinant of I is equal to

If m = 2 and n = 5, then p equals to

Which of the following ordered triplet ( m, n, p ) is false?

The number of all possible values of θ, where 0 < θ < π, for which the system of equations(y + z)cosθ = (xyz)sinθxsinθ = 2cos3θ/y + 2sin3θ/z(xyz)sinθ = (y + 2z)cosθ + ysin3θhave a solution (x0, y0, z0) with y0z0 ≠ 0, is

Let $M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}$ and $adjM = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?(A) $a + b = 3$(B) $\det(adjM^2) = 81$(C) $(adjM)^{-1} + adjM^{-1} = -M$(D) If $M \begin{bmatrix} \alpha \\ 1 \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $\alpha - \beta + \gamma = 3$

The determinant xxz+xzyzzxx+y equals -

The number of real values of x satisfying is -

The number of real values of x satisfying <mfenced open="|

Let $f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1+\sin 2x \end{vmatrix}, x \in \left[ \frac{\pi}{6}, \frac{\pi}{3} \right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

Let a, λ, μ ∈ R. Consider the system of linear equationsax + 2y = λ3x - 2y = μWhich of the following statement(s) is(are) correct?(A) if a = -3, then the system has infinitely many solutions for all values of λ and μ(B) if a ≠ -3, then the system has a a unique solution for all values of λ and μ(C) if λ + μ = 0, then the system has infinitely many solutions for a = -3(D) if λ + μ ≠ 0, then the system has no solution for a = -3

For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?(A) NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric(B) MN − NM is skew symmetric for all symmetric matrices M and N(C) MN is symmetric for all symmetric matrices M and N(D) (adjM)(adjN) = adj(MN) for all invertible matrices M and N

Let A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is :-

If the system of equation x + (\sqrt{2}\sin\alpha)y + (\sqrt{2}\cos\alpha)z = 0x + (\cos\alpha)y + (\sin\alpha)z = 0x + (\sin\alpha)y - (\cos\alpha)z = 0has a non-trivial solution, then \alpha =

The number of real values of x satisfying is -

If ab+abbc+bc1bc+bcca+ca1ca+caab+ab1=0, where a, b, c ∈ R+, then which of the following is necessarily true -

Let the system of linear equations4x + λy + 2z = 02x - y + z = 0μx + 2y + 3z = 0, λ, μ ∈ Rhas a non-trivial solution. Then which of the following is true ? [JEE (Main) 2021]

Consider the system of equations : x + ay = 0, y + az = 0 and z + ax = 0. Then the set of all real values of 'a' for which the system has a unique solution is :

If $a, b, c$ are the roots of the equation $x^3 + 2x^2 + 1 = 0$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} =$

If the system of equation, a2x - ay = 1 - a & bx + (3 - 2b)y = 3 + a possess a unique solution x = 1, y = 1 than :

For the matrix $A = \begin{bmatrix} 4 & -4 & 5 \\ -2 & 3 & -3 \\ 3 & -3 & 4 \end{bmatrix}$ find $A^{-2}$.

Which of the following is(are) NOT the square of a 3 × 3 matrix with real entries?(A) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (B) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (C) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (D) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$

If \(\det(A)\)=4, then det(A\)^4A^2A⁻^5A^{-1}) equals:

If Δ(x) = 02x-22x+8x-14x2+700x+4 and f(x) = ∑i=13∑j=13aijcij, where a_{ij} is the element of i^th row and j^th column in Δ(x) and c_{ij} is the cofactor of a_{ij} ∀ i and j, then find the greatest value of f(x), where x ∈ [-3, 18].

Let A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}, a, b \in \mathbb{R}. If for some n \in \mathbb{N}, A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix} then n + a + b is equal to.

If P is a 3 × 3 real matrix such that PT = aP + (a-1)I, where a > 1, then

For a matrix $A = \begin{bmatrix} 1 & 2r-1 \\ 0 & 1 \end{bmatrix}$, the value of $\prod_{r=1}^{50} \begin{bmatrix} 1 & 2r-1 \\ 0 & 1 \end{bmatrix}$ is equal to -

Let α ∈ (0, ∞) and A = 12α101012. If det(adj(2A - Aᵀ).adj(A - 2Aᵀ)) = 2⁸, then (det(A))² is equal to:

If Δ(x) = <mfenced close="|

Let S = {√n : 1 ≤ n ≤ 50 and n is odd}. Let a ∈ S and A = 10a-110-a01. If Σa∈S det(adj A) = 100λ, then λ is equal to

If the system of equationsx + y + z = 62x + 5y + az = bx + 2y + 3z = 14has infinitely many solutions, then a + b is equal to :

If \(A_1, A_2, \ldots, A_{2n-1}\) are \(n\) skew-symmetric matrices of same order, then \(B = \displaystyle\sum_{r=1}^{n}(2r-1)(A_{2r-1})^{2r-1}\) will be

Let A = \begin{pmatrix} 2 & 3 \\ a & 0 \end{pmatrix}, a \in \mathbb{R} be written as P + Q where P is a symmetric matrix and Q is skew symmetric matrix. If \det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to:

If A & B are square matrices of order 2 such that A + adj(BT) = 2112 & AT - adj(B) = 0110,then-(A) B is symmetric matrix(B) An = A ∀ n ∈ N(C) |A + A2 + A3 + A4 + A5| = 0(D) |B + B2 + B3 + B4 + B5| = 0

For positive numbers x, y and z, the numerical value of the determinant is -