Determinants Questions (2072)

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 -

Let P be a matrix of order 3 x 3 such that all the entries in P are from the set {-1, 0, 1}. Then, the maximum possible value of the determinant of P is ____

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Let α, β and γ be real numbers. consider the following system of linear equationsx + 2y + z = 7x + αz = 112x - 3y + βz = γMatch each entry in List-I to the correct entries in List-IIList-IList-II(P) If β = 1/2(7α - 3) and γ = 28, then the system has(1) a unique solution(Q) If β = 1/2(7α - 3) and γ ≠ 28, then the system has(2) no solution(R) If β ≠ 1/2(7α - 3) where α = 1 and γ ≠ 28, then the system has(3) infinitely many solutions(S) If β ≠ 1/2(7α - 3) where α = 1 and γ = 28, then the system has(4) x = 11, y = -2 and z = 0 as a solution(5) x = -15, y = 4 and z = 0 as a solution

If \[f(x) = \begin{vmatrix} 1 & x & \frac{x^2}{2} \\ 0 & 2 & x \\ 0 & 2 & 6x \end{vmatrix}\], then \(f'(x)\) is equal to

If \(\ell will always be greater than -

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 -

Let d ∈ R, and A = -24+dsinθ-21sinθ+2d52sinθ-d-sinθ+2+2d, θ ∈ [0, 2π]. If the minimum value of det(A) is 8, then a value of d is :

If A and B are two orthogonal matrices of order 3, then -(A) A and B both will be invertible matrices(B) matrix ABA will also be orthogonal(C) matrix A^2B^2 will also be orthogonal(D) maximum value of det\left(\frac{A}{2} adj(2B)\right) is 8.

Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is

The number of triplets (α, β, γ) satisfying the following constraints2α - β + 3γ = 4α + β - 3γ = -15α - β + 3γ = 7αβγ ≤ 0& α, β, γ ∈ I

Let a determinant is given by A = abcpqrxyz and suppose A = 6. If B = p+xq+yr+za+xb+yc+za+pb+qc+r then

If A = ete-t cos te-t sin tet-e-t cos t - e-t sin t-e-t sin t + e-t cos tet2e-t sin t-2e-t cos t Then A is -

For the matrices $A=\begin{bmatrix}3&-4\\1&-1\end{bmatrix}$ and $B=\begin{bmatrix}-29&49\\-13&18\end{bmatrix}$, if $(A^{15}+B)\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$, then among the following which one is true?

If A-1 = 1-100-2100-1, then

Let $|A|=6$, where $A$ is a $3\times3$ matrix. If $|\text{adj}(3\,\text{adj}(A^2\cdot\text{adj}(2A)))|=2^m\cdot3^n$, $m,n\in\mathbb{N}$, then $m+n$ is equal to _____.

If $A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$, $A^{-1} = \alpha A + \beta I$ and $\alpha + \beta = -2$, then $4\alpha^2 + \beta^2 + \lambda^2$ is equal to:

Let f(x) = 1x-1x2(x-1)(x-1)(x-2)x(x-1)3(x-1)(x-2)(x-1)(x-2)(x-3)(x-1)(x-2) & Dr = 1102r70173r+1111. The value of f(50) - D5 is -