Determinants Questions - JEE

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 -

Let $A$ be a $2\times2$ symmetric matrix such that $A\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\7\end{bmatrix}$ and the determinant of $A$ be 1. If $A^{-1}=\alpha A+\beta I$, where $I$ is an identity matrix of order $2\times2$, then $\alpha+\beta$ equals

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

If the matrix $A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$ where $a, b, c$ are positive real numbers such that $abc = 1$ and $A^T A = I$, then find the value of $a^3 + b^3 + c^3$.

Let A = $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{bmatrix}$$. Then the sum of the diagonal elements of the matrix $$(A + I)^{11}$$ is equal to:

Let P be a 2 x 2 matrix such that P\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} and P^2\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}. If p_1 and p_2 (p_1 > p_2) are two values of p for which det(P - pI) = 0, where I is an identity matrix of order 2, then (5p_1 + 2p_2) is equal to [Note : det(M) denotes determinant of square matrix M]

If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then xzy2 is equal to :

Let P be a 2 x 2 matrix such that P\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \end{bmatrix} and P^2\begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}. If p_1 and p_2 (p_1 > p_2) are two values of p for which det(P - pI) = 0, where I is an identity matrix of order 2, then (5p_1 + 2p_2) is equal to [Note : det(M) denotes determinant of square matrix M]

If $\det(A)$=5, then det(A\)^{-1}$A$^TA^{-1}) equals:

If $\det(A)$=3, then det(A\)^2A^{-1}) equals:

The determinant <mfenced close="|

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(x-1)(x-2) & Dr=1102r70173r+1111. The value of ∑r=110Dr is

System of linear equations in x, y, z have infinite solutions which2x + y + z = 1x - 2y + z = 23x - y + 2z = 3

If $\det(A)$=k, then det(A\)⁻^2A^3) equals:

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

If $\det(A)$=5, then det((A\)^{-1})^2) equals:

Let $\alpha\beta\neq0$ and $A=\begin{bmatrix}\beta&\alpha&3\\\alpha&\alpha&\beta\\-\beta&\alpha&2\alpha\end{bmatrix}$. If $B=\begin{bmatrix}3\alpha&-9&3\alpha\\-\alpha&7&-2\alpha\\-2\alpha&5&-2\beta\end{bmatrix}$ is the matrix of cofactors of the elements of $A$, then $\det(AB)$ 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 $A=\begin{bmatrix}2&a&0\\1&3&1\\0&5&b\end{bmatrix}$. If $A^3=4A^2-A-21I$, where $I$ is the identity matrix of order $3\times3$, then $2a+3b$ is equal to:

If $\det(A)$=3, then det(A\)^5A⁻^3A^{-1}) equals:

If \det(A)=k, then $\det(kA)$ for $2 \times 2$ matrix is:

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

If $\det(A)$=5, then det(A\)⁻^2A^4A^{-1}) equals:

If $\det(A)$=2, then det(A\)^2A^3A⁻^4) equals:

If A is a non-singular matrix and AT denotes the transpose of A, then :

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

If $\det(A)$=4, then det(A\)^4A^2A⁻^5) equals:

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + A^{19} + ... + A + I = ?

If $\det(A)$=3, then det(A\)^3A^2A⁻^4) equals:

If $A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$ and $(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, then $13(a^2 + b^2)$ is equal to ____.

Determinants Questions (2072)