Determinants Questions (2072)

If the product of n matrices $\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \dots \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is equal to the matrix $\begin{bmatrix} 1 & 378 \\ 0 & 1 \end{bmatrix}$ then the value of n is equal to -

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-

If the system of linear equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ has infinitely many solutions, then

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 number of all possible values of θ, where 0 < θ < π, for which the system of equations(y + z)cosθ = (xyz)sinθxsinθ = 2cos3θ/y + 2sin3θ/z(xyz)sinθ = (y + 2z)cosθ + ysin3θhave a solution (x0, y0, z0) with y0z0 ≠ 0, is

Match the following: A -> S; B -> R; C -> P; D -> Q

Let a - 2b + c = 1. If f(x) = <mfenced open="|

For α, β ∈ R and a natural number n, let A_r = <mfenced open="|

If the determinant <mfenced open="|

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

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19}(A-I) + A^{18}(A-I)^2 + \dots + (A-I)^{20} is equal to

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is (are) TRUE?

If \det(A) = 2 and \det(B) = 3, then \(\det(A\)\)^{-1}\(B\)^2) is:

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

If \(A = [a_{ij}]_{n \times n}\) and \(a_{ij} = (i^2 + j^2 - ij)(j - i)\), where \(n\) is odd, then the value of \(tr.(A)\) is equal to:

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay, then a^2 + b^2 + c^2 + 2abc is equal to

If \det(A)=4 and \det(B)=5, then \(\det((AB)\)^{-1}) equals:

If \det(A)=2, then \(\det(A\)\)^{-1} + \(I) equals:

For positive numbers x, y and z, the numerical value of the determinant 1logxylogxzlogyx1logyzlogzxlogzy1 is -

If \(\det(A)\)=3, then det(A\)^{-1}\(A\)^{-1}) equals:

Which of following statement is/are false -(A) t is divisible by (α - β)(B) t is divisible by (β - γ)(C) t is divisible by (γ - α)(D) (γ - α) is divisible by t

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