Determinants Questions (2072)

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?

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

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices : Tp = {A = abca : a, b, c ∈ {0, 1, 2, ..., p-1}}. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is -

Let A = 2-111. If the sum of the diagonal elements of A13 is 3n, then n is equal to ____.

If px^4 + qx^3 + rx^2 + sx + t = \[ \begin{vmatrix} \(x^2+3x\) & \(x-1\) & \(x+3\) \\ \(x+1\) & \(2-x\) & \(x-3\) \\ \(x-3\) & \(x+4\) & 3x \end{vmatrix} \], then t is equal to:

The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is :-

Let S = {A = 01c1ad1be : a, b, c, d, e ∈ {0, 1} and |A| ∈ {-1, 1}}, where |A| denotes the determinant of A. Then the number of elements in S 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.

If A and B are symmetric matrices and AB = BA, then A-1B is a -

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 :

The system of linear equations x + y + z = 6, x + 2y + 3z = 14 and 2x + 5y + pz = q have -

If A is a symmetric and B skew symmetric matrix and A + B is non singular and C = (A + B)-1(A - B) then CT AC =

The value of the determinant of a matrix is given by the expression. If the determinant is 1, what is the value?

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

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

If A = 1000110-24, I = 100010001 and A-1 = 16(A2 + cA + dI), then the value of c + d is:

If px4 + qx3 + rx2 + sx + t = x2+3xx-1x+3x+12-xx-3x-3x+43x then t is equal to -

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

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 =