Determinants Questions - JEE

Consider three matrices $A = \begin{bmatrix} 2 & 1 \\ 4 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}$ and $C = \begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}$, then the value of the sum $tr(A) + tr\left(\frac{ABC}{2}\right) + tr\left(\frac{A(BC)^2}{4}\right) + tr\left(\frac{A(BC)^3}{8}\right) + \ldots \infty$ is

If A is order 3 square matrix such that $|A| = 2$, then $|\text{adj}(\text{adj}(\text{adj } A))|$ is

If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A+Bx)(x-A)^2$ then $A + 2B$ equals

If $A = \begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}$ and $B$ is the adjoint of $A$, then $\det(AB+2I)$ is (where $I$ is $3\times3$ identity)

Let $k$ be a positive real number and let $A = \begin{bmatrix} 2k & 2\sqrt{k} & 2\sqrt{k} \\ 2\sqrt{k} & 1 & -2k \\ -2\sqrt{k} & 2k & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k-1 & \sqrt{k} \\ 1-2k & 0 & 2\sqrt{k} \\ -\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix}$. If $\det(\text{adj }A) + \det(\text{adj }B) = 10^6$, then greatest integer of $k$ is equal to

Let $M$ be a $3 \times 3$ matrix satisfying $$M \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -1 \\ 2 \\ 3 \end{bmatrix}, M \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix}$$ and $$M \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 12 \end{bmatrix}$$. Then the sum of diagonal entries of $M$ is ____

Let $\omega$ be the complex number $\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}$. Then, the number of distinct complex numbers $z$ satisfying $$\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 0$$ is equal to ____

The system $(1-\lambda)x+3y-4z=0$, $x-(3+\lambda)y+5z=0$, $3x+y-\lambda z=0$ possesses non-trivial solutions for

The characteristic equation of a matrix $A$ is $\lambda^3 - 5\lambda^2 - 3\lambda + 2 = 0$ then $|\text{adj}A| = $

For what values of x the matrix \[\begin{bmatrix} 3+x & 5 & 2 \\ 1 & 7+x & 6 \\ 2 & 5 & 3+x \end{bmatrix}\] has the rank 2?

Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\dfrac{q_{31} + q_{32}}{q_{21}}$ equals

If $x_1, x_2, x_3, \ldots, x_{13}$ are in A.P then the value of $$\begin{vmatrix} e^{x_1} & e^{x_4} & e^{x_7} \\ e^{x_4} & e^{x_7} & e^{x_{10}} \\ e^{x_7} & e^{x_{10}} & e^{x_{13}} \end{vmatrix}$$ is ____

If $a_i^2 + b_i^2 + c_i^2 = 1, (i = 1, 2, 3)$ and $a_i a_j + b_i b_j + c_i c_j = 0$ ($i \neq j; i, j = 1, 2, 3$) then the value of $$\begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$$ is ____

If $\det(A)$=5, then det(A\)^7A⁻^5A^{-1}) equals:

If $\det(A)$=3, then det(A\)⁻^2A^6A⁻^3A) equals:

If $\det(A)$=k, then det(A\)^{-1}$A$^nA^nA⁻^n) equals:

If $\det(A)$=4, then det(A\)⁻^3A^5A^{-1}) equals:

If $\det(A)$=3, then det(A\)^4A⁻^2A^{-1}) equals:

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

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

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

If $\det(A)$=4, then det(A\)^{-1}$A$^{-1}$A$^3) equals:

The rank of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ \lambda & 2 & 4 \\ 2 & -3 & 1 \end{bmatrix}$ is 3 if

If $\text{adj } B = A$, $|P| = |Q| = 1$, then $\text{adj}(Q^{-1} B P^{-1})$ is

The number of values of k, for which the system of equations:$(k+1)x + 8y = 4k$$kx + (k+3)y = 3k - 1$has no solution, is

If $A = \begin{bmatrix} a & b & c \\ x & y & z \\ p & q & r \end{bmatrix}$, $B = \begin{bmatrix} q & -b & y \\ -p & a & -x \\ r & -c & z \end{bmatrix}$ and if A is invertible, then which of the following is not true?

If $\det(A)$=k, then det(A\)^n(A\)⁻^n + \(I)^p) equals:

For the equation $\begin{vmatrix} 1 & x & x^2 \\ x^2 & 1 & x \\ x & x^2 & 1 \end{vmatrix} = 0$,

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

Evaluate the cyclic determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$.

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

If $\det(A)$=3, then det(A\)^5A^2A⁻^2A⁻^5A^3) equals:

If $A^2 - A + I = 0$, then the inverse of $A$ is

If $\det(A)$=4, then det(A\)^6A^3A⁻^3A⁻^6A^4A⁻^4) equals:

Let $A$ and $B$ be two square matrices of the same size such that $AB^T + BA^T = O$. If $A$ is a skew-symmetric matrix then $BA$ is

Given $A = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$. If $A^{-50}$ is evaluated at $\theta = \dfrac{\pi}{12}$, find $A^{-50}$.

If $\det(A)$=5, then det((A\)^6A⁻^6)^7) equals:

If $\det(A)$=3, then det((A\)^4A⁻^4)^5) equals:

The number of values of $k$ for which the linear equations:$4x + ky + 2z = 0$$kx + 4y + z = 0$$2x + 2y + z = 0$possess a non-zero solution is

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

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

If $\det(A)$=5, then det(A\)^7A^4A⁻^4A⁻^7A^5) equals:

If $\det(A)$=k, then det(A\)^nA^mA⁻^m) equals:

If $\det(A)$=5, then det(A\)^6(A\)⁻^6 + \(I)) equals:

If $\det(A)$=k, then det(A\)^n(A\)⁻^n + \(I)) equals:

If $\det(A)$=k, then det((A\)^nA⁻^n)^m) equals:

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$, $a, b \in N$. Then

If $\det(A)$=k, then det(A\)^n(A\)⁻^n - \(I)) equals:

If $A$ is square matrix of order 3 then $|\text{adj}(\text{adj } A)| =$

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

Determinants Questions (2072)