Matrices & Determinants Questions (2045)

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 -

If determinant has all elements same in a row, its value is:

If \(A\) is a \(2 \times 2\) matrix, then adj(adj \(A) equals:

22. D = 104+2107+3108+8109+9102+8103-4103-5108+b106+a where a, b, both ∈ {1,2,3,4,5,6,7,8,9}Number of ordered pairs (a, b) such that D = 2n + 1, n ∈ Z is

If \[A = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\]then the matrix \(A^{-50}\) is equal to (JEE Main 2019)

Let p, q, r be nonzero real numbers that are, respectively, the 10th, 100th and 1000th terms of a harmonic progression. Consider the system of linear equationsx + y + z = 110x + 100y + 1000z = 0qrx + pry + pqz = 0Match List-I with List-II.

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M-1 = adj(adj M), then which of the following statement is/are ALWAYS TRUE?

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19} + ... + A + I = ?

Let P = \begin{pmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{pmatrix} and Q = [q_{ij}] be two 3×3 matrices such that Q - P^{-1} = I_3. Then, \frac{5(q_{21} + q_{31})}{q_{32}} is equal to

If $S_r = \begin{vmatrix} 2r & x & n(n+1) \\ 6r^2-1 & y & n^2(2n+3) \\ 4r^3-2nr & z & n^3(n+1) \end{vmatrix}$, then $\sum_{r=1}^n S_r$ does not depend on -

Matrix A such that A^2 = 2A - I, where I is the identity matrix. Then, for n \geq 2, A^n is equal to

Let A be a non-singular matrix of order 3. If det(3adj(2adj((detA)A))) = 3-13 ⋅ 2-10 and det(3adj(2A)) = 2m ⋅ 3n, then |3m + 2n| is equal to ____.

Solve for x if the determinant \[\begin{vmatrix} t-1 & 3t+1 & 2t \\ t-1 & 4t-2 & t+3 \\ 2 & 3t+1 & 3(t-1) \end{vmatrix} = 0\]

Let Dk = <mfenced open="|

If Δ = a1b1c1a2b2c2a3b3c3 and A2, B2, C2 are respectively cofactors of a2, b2, c2 then a1A2 + b1B2 + c1C2 is equal to -

Let M = \begin{bmatrix} a & -360 \\ b & c \end{bmatrix}, where a, b and c are integers. Find the smallest positive value of b such that M^2 = \mathbf{0}, where \mathbf{0} denotes 2 \times 2 null matrix.

If the system of equations x + y + z = 5, x + 2y + 3z = 9, x + 3y + az = b has infinitely many solutions, then b - a equals

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

If the system of linear equations x - 4y + 7z = g 3y - 5z = h -2x + 5y - 9z = k is consistent, then :

A value of θ ∈ (0, π/3), for which\[\begin{vmatrix} 1 + \cos^2 θ & \sin^2 θ & 4\cos 6θ \\ \cos 2θ & 1 + \sin^2 θ & 4\cos 6θ \\ \cos 2θ & \sin^2 θ & 1 + 4\cos 6θ \end{vmatrix} = 0\]

Let \(f(x) = \begin{vmatrix} 1 & 2x & (x-1) \\ 3x(x-1) & (x-1)(x-2) & x(x-1) \\ 1 & 2x & x-1 \\ 3x & x-2 & x \end{vmatrix}\)Find \(f(50)\).

If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3 + px + q = 0\), then the value of the determinant \(\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix}\) is

If the determinant a+pl+xu+fb+qm+yv+gc+rn+zw+h splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K, is-

Given system of linear equations:\(x + y + z = 5\) …(i)\(x + 2y + 2z = 6\) …(ii)\(x + 3y + \lambda z = m\) …(iii)where \(\lambda, m \in \mathbb{R}\)If the above system has infinitely many solutions, find \(\lambda + m\).

If a, b, c are in AP, then x+1x+2x+ax+2x+3x+bx+3x+4x+c equals -

Find k such that the following system of equations has infinitely many solutions:3x - 2y - kz = 02x - 4y - 2z = 0x + 2y - z = 0

The system of linear equations x + λy - z = 0, λx - y - z = 0, x + y - λz = 0 has a non-trivial solution for :

If α, β, γ satisfy the equation x111x111x=0, then

There are two numbers x making the value of the determinant 1-252x-1042x equal to 86. The sum of these two numbers, is-

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

System of linear equations in x, y, z have infinite solutions which2x + y + z = 1x - 2y + z = 23x - y + 2z = 3(A) can be written as (-3λ -1, λ, 5λ + 3) ∀ λ ∈ R(B) can be written as (3λ -1, -λ, -5λ + 3) ∀ λ ∈ R(C) are such that every solution satisfy x - 3y + 1 = 0(D) are such that none of them satisfy 5x + 3z = 1

If a1+b1xa1x+b1c1a2+b2xa2x+b2c2a3+b3xa3x+b3c3=0, then possible conditions is/are -

The value of α(β² + γ²) + β(γ² + α²) + γ(α² + β²) is divisible by -