Matrices & Determinants Questions (2045)

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

The total number of distinct x ∈ R for which <mfenced open="|

Let A, B, C, D be real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix M = ABCD, then M2016 is equal to

The determinant a2a2-(b-c)2bcb2b2-(c-a)2cac2c2-(a-b)2ab is divisible by -

The total number of matrices A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which AT A = 3I3 is :-

If a^2 + b^2 + c^2 = -2 and f(x) = 1+a2x(1+b2)x(1+c2)x(1+a2)x1+b2x(1+c2)x(1+a2)x(1+b2)x1+c2x then f(x) is a polynomial of degree-

Consider a matrix A(θ) = sinθcosθ-cosθsinθ then

If sin2xcos2x4sin2x2tan2x2cos2x-sin2x-2cos4xtan2x2sin4x = a_0 + a_1(cos x) + a_2(cos^2 x) + ........ + a_n(cos^n x), then a_0 is -

If the system of equations x + y + z = 6 2x + 5y + αz = β x + 2y + 3z = 14 has infinitely many solutions, then α + β is equal to :

Let A + 2B = $\begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and 2A - B = $\begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$, then Tr(A) - Tr(B) has the value equal to

Let the determinant of a square matrix A of order m be m - n, where m and n satisfy 4m + n = 22 and 17m + 4n = 93. If det (n adj (adj (mA))) = 3a 5b 6c. Then a + b + c is equal to:

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 is skew symmetric matrix of order 3 and X be another matrix of same order, then |XA + AXT| is (where |P| denotes determinant of matrix P) -

22. $D = \begin{vmatrix} 10^4 + 2 & 10^7 + 3 & 10^8 + 8 \\ 10^9 + 9 & 10^2 + 8 & 10^3 - 4 \\ 10^3 - 5 & 10^8 + b & 10^6 + a \end{vmatrix}$ where $a, b$, both $\in \{1,2,3,4,5,6,7,8,9\}$Number of ordered pairs $(a, b)$ such that $D = 2n + 1, n \in Z$ is

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

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 $D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}$ and $D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}$ then the value of $\frac{D_1}{D_2}$ where $b \neq 0$ and $ad \neq bc$, is

Let $D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}$ and $D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}$ then the value of $\frac{D_1}{D_2}$ where $b \neq 0$ and $ad \neq bc$, is

Let A be a 3x3 matrix such that A^2 = A. If det(A) = 0 and the trace of A is 2, then the rank of A is:

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

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

Matrix A = x321y422z, if xyz = 60 and 8x + 4y + 3z = 20, then A(adj A) is equal to -

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

Let M = \begin{pmatrix} \sin^2 \theta & -1-\sin^2 \theta \\ 1+\cos^2 \theta & \cos^2 \theta \end{pmatrix} = \alpha I + \beta M^{-1}, where \alpha = \alpha(\theta) and \beta = \beta(\theta) are real number, and I is the 2 \times 2 identity matrix. If \alpha^* is the minimum of the set \{\alpha(\theta): \theta \in [0, 2\pi]\} and \beta^* is the minimum of the set \{\beta(\theta): \theta \in [0, 2\pi]\}, then the value of \alpha^* + \beta^* is

If <mfenced open="|

If A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} and A^2 - kA - I_2 = 0, then value of k is-

If A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} and det. (A^n - I) = 1 - \lambda^n, n \in N then the value of \lambda, is -

If $\begin{vmatrix}\sin x+1&\sin 2x&\sin 3x\\ \sin 2x&\sin 3x+a&\sin 4x\\ \sin 3x&\sin 4x&\sin 5x+a^2\end{vmatrix}=2025(f(x)+45)$ where $f(x)$ is a function of $x$ and $a$ is complex, then sum of all possible values of $a$ is

Let $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B = I + \text{adj}(A) + (\text{adj } A)^2 + \dots + (\text{adj } A)^{10}$. Then, the sum of all the elements of the matrix $B$ is :

Let f(x) = 1+sin2xcos2x4sin2xsin2x1+cos2x4sin2xsin2xcos2x1+4sin2x, then the maximum value of f(x), is-

For a determinant Δ of order 3, the element aij is defined as aij = tan-1(tan(i - j)) ∀ i, j, then the value of Δ is equal to (where 'i' represents row and 'j' represents column)

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19}(\text{adj } A) + \dots + A(\text{adj } A)^{19} + (\text{adj } A)^{20} \text{ is equal to}

Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ²z = μ² + 15 where λ, μ ∈ R. Which one of the following statements is NOT correct?