Determinants Questions (2072)

Let P = , A = 1101 and Q = PAPT. If PTQ2007P = abcd, then 2a + b - 3c - 4d equal to

Which of the following options is/are correct?(A) Let $A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ are two matrices such that $AB = (AB)^{-1}$ and $AB \neq I$ (where $I$ is an identity matrix of order $3 \times 3$). Then $tr\left(AB + (AB)^2 + (AB)^3 + ..... + (AB)^{100}\right)$ equals 100.(B) If $A$ and $B$ are square matrices of order 3, where $|A| = -2$ and $|B| = 1$, then $|(A^{-1})adj(B^{-1})adj(2A^{-1})|$ equals 8.(C) If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $F(x).F(y) = F(x + y)$ and $[F(x)]^{-1} = F(-x)$.(D) Let $X$ be the solution set of the equation $A^x = I$, where $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$ and $I$ is the corresponding unit matrix and $X \subseteq \mathbb{N}$ then the minimum value of $\sum(\cos^n \theta + \sin^n \theta), \theta \in \mathbb{R}, n \in \mathbb{Z}$ is 2.

Let A = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} and B = \begin{pmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & -14^2 \\ -15^2 & 16^2 & 17^2 \end{pmatrix}, then the value of A'BA is:

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

If a2 + b2 + c2 = -2 and f(x) = <mfenced open="|

The number of θ ∈ (0, 4π) for which the system of linear equations 3(sin 3θ)x - y + z = 2 3(cos 2θ)x + 4y + 3z = 3 6x + 7y + 7z = 9 has no solution is :

Let A be a 2 x 2 matrix with non-zero entries and let A2 = I, where I is 2 x 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement-1 : Tr(A) = 0.Statement-2 : |A| = 1.(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.(C) Statement-1 is true, Statement-2 is false.(D) Statement-1 is false, Statement-2 is true.

Let f(x) = |1+sin^2 xcos^2 x4sin 2x||sin^2 x1+cos^2 x4sin 2x||sin^2 xcos^2 x1+4sin 2x|, then the maximum value of f(x), is-

If $A$ and $B$ are square matrices of order 3 such that $4A^T = 3B$ and $2AB^T = 3A^T B$, then the value of $\frac{|A|^2}{|B|}$ is equal to

If α, β ≠ 0, and f(n) = α^n + β^n and , then K is equal to :

Let a, b, c, l, m, n ∈ R such that al + bm + cn = 0, bl + cm + an = 0, cl + am + bn = 0. If a, b & c are distinct & f(x) = ax^3 + bx^2 + cx + 5, then the value of f(1) is

The value of an odd order determinant in which aij + aji = 0 ∀ i, j is -

If a, b, c > 0 and x, y, z ∈ R, then the determinant is equal to -

Let A and B be two invertible matrices of order 3 x 3. If det(ABA^T) = 8 and det(AB^-1) = 8, then det(BA^-1 B^T) is equal to :-

If a, b, c are sides of a scalene triangle, then the value of <mfenced close="|

If A = 1234, then A2 - 5A - 2I is equal to

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

Let M be a 3 x 3 non-singular matrix with det(M) = 4. If M-1 adj(adj M) = k2I, then the value of 'k' may 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:

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 -

Find the number of A = [aij]2x2 satisfying aij is 1 or -1 and a11a21 + a12a22 = 0.

Let A = $\begin{bmatrix} l-3 & a & b \\ c & 6 & d \\ e & f & 9-l \end{bmatrix}$, B = adj(A) and C = adj(B). If |A| = 5, then tr(C) is (where |X|, tr(X) & adj(X) denote determinant value, trace and adjoint of matrix X respectively) -

Number of real values of λ for which the matrix A = $\begin{bmatrix} \lambda-1 & \lambda & \lambda+1 \\ 2 & -1 & 3 \\ \lambda+3 & \lambda-2 & \lambda+7 \end{bmatrix}$ has no inverse

If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :

Let a1, a2, a3, ..., a10 be in G.P. with ai > 0 for i = 1,2,..., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which . Then the number of elements in S, is :

The determinant (x2+1)2(xy+1)2(xz+1)2(xy+1)2(y2+1)2(yz+1)2(xz+1)2(yz+1)2(z2+1)2 = k(x-y)^2(y-z)^2(z-x)^2, then k =

If the system of equations x + y - 3 = 0, (1 + K)x + (2 + K)y - 8 = 0 & x - (1 + K)y + (2 + K) = 0 is consistent then the value of K may be -

Which of the following values of α satisfy the equation (1+α)2(1+2α)2(1+3α)2(2+α)2(2+2α)2(2+3α)2(3+α)2(3+2α)2(3+3α)2 = -648α?

If a, b, c are in A.P. and α, β, γ are positive real numbers in G.P., then the equation x+ax2+logαkx+bx2+logβkx+cx2+logγk = 0 :-

Let A be a 3x3 matrix such that A2 - 5A + 7I = 0. If An = 5n A - 7n I for some n, then n is equal to:

If A is a square matrix of order 3 such that det(A) = 3 and det(adj(-4 adj(-3 adj(3 adj((2A)^(-1)))))) = 2^m 3^n, then m + 2n is equal to:

Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if(A) the first column of M is the transpose of the second row of M(B) the second row of M is the transpose of the first column of M(C) M is a diagonal matrix with nonzero entries in the main diagonal(D) the product of entries in the main diagonal is not the square of an integer

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

The determinant , then k =

If A and B are square matrices of order 3, then the true statement is/are (where I is unit matrix).(A) det (-A) = -det A(B) If AB is singular then atleast one of A or B is singular(C) det (A + I) = 1 + det A(D) det (2A) = 2^3 det A

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 :

3. If px^4 + qx^3 + rx^2 + sx + t = x2+3xx-1x+3x+12-xx-3x-3x+43x then t is equal to -

The value of θ lying between -π/4 & π/2 and 0 ≤ A ≤ π/2 and satisfying the equation 1+sin2 Acos2 A2sin 4θsin2 A1+cos2 A2sin 4θsin2 Acos2 A1+2sin 4θ = 0 are -

Let B = $$\begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix}$$ and A be a 2 x 2 matrix such that AB^{-1} = A^{-1}. If BCB^{-1} = A and C^4 + \alpha C^2 + \beta I = 0, then 2\beta - \alpha is equal to :

Let A = 100210321. If u1 and u2 are column matrices such that Au1 = 100 and Au2 = 010, then u1 + u2 is equal to:

If A and B are symmetric matrices of the same order and X = AB + BA, Y = AB - BA then (XY)T is equal to -

If the equations a(y + z) = x, b(z + x) = y, c(x + y) = z (where a, b, c ≠ -1) have nontrivial solutions, then find the value of 1/(1+a) + 1/(1+b) + 1/(1+c).

Consider the following statementsStatement-1 : If A is an idempotent non-zero matrix and I is an identity matrix of the same order, such that (A + I)n = I + 127 A. (n ∈ N), then 'n' has 3 positive divisors.Statement-2 : Let A = 3x216x, B = [a b c] and C = (x+2)25x22x5x22x(x+2)22x(x+2)25x2 be three given matrices, where a, b, c and x ∈ R. Given that tr(AB) = tr(C) ∀ x ∈ R, where tr(A) denotes trace of A. Solving, we get a + b + c = 7.Then, which of the following options is/are correct ?

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

For which of the following ordered pairs (μ, δ), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = μ 4x + 4y + 4z = δ is inconsistent?

If the adjoint of a 3 × 3 matrix P is 144217113, then the possible value(s) of the determinant of P is (are) -

Match the following for the system of linear equations λx + y + z = 1, x + λy + z = λ, x + y + λz = λ2. Column-IColumn-II(A) λ = 1(P) unique solution(B) λ ≠ 1(Q) infinite solutions(C) λ ≠ 1, λ ≠ -2(R) no solution(D) λ = -2(S) finite many solutions

Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:

If x, y, z are distinct digits (0 ≤ x, y, z ≤ 9) & the minimum possible value of z9yxzy9x9zyx is λ then λ83700 is (where 9x, 9y & 9z are two digits number)

Let P = 3-1-220α3-50, where α ∈ R. Suppose Q = [qij] is a matrix such that PQ = kI, where k ∈ R, k ≠ 0 and I is the identity matrix of order 3. If q23 = -k/8 and det(Q) = k2/2, then(A) α = 0, k = 8(B) 4α - k + 8 = 0(C) det(P adj(Q)) = 29(D) det(Q adj(P)) = 213