Determinants Questions - JEE

Evaluate the determinant $\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix}$.

If $\det(A)$=5, then det(A\)^6(A\)⁻^6 + \(I)^2) equals:

Let $A$ be a $2 \times 2$ matrix with non-zero entries and let $A^2 = I$, where $I$ is $2 \times 2$ identity matrix. Define tr$(A)$ = sum of diagonal elements of $A$ and $|A|$ = determinant of matrix $A$.Statement-1: tr$(A) = 0$Statement-2: $|A| = 1$

If $pqr \neq 0$ and the system of equations \[(p+a)x + by + cz = 0\] \[ax + (q+b)y + cz = 0\] \[ax + by + (r+c)z = 0\] has a non-trivial solution, then value of $\dfrac{a}{p} + \dfrac{b}{q} + \dfrac{c}{r}$ is

If $X=\begin{bmatrix}x\\y\\z\end{bmatrix}$ is a solution of the system of equations $AX=B$, where $\text{adj}\,A=\begin{bmatrix}4&2&2\\-5&0&5\\1&-2&3\end{bmatrix}$ and $B=\begin{bmatrix}4\\0\\2\end{bmatrix}$, then $|x+y+z|$ is equal to:

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

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

Consider the matrices \[A = \begin{bmatrix} 4 & 6 & -1 \\ 3 & 0 & 2 \\ 1 & -2 & 5 \end{bmatrix},\quad B = \begin{bmatrix} 2 & 4 \\ 0 & 1 \\ -1 & 2 \end{bmatrix},\quad C = \begin{bmatrix} 3 \\ 1 \\ 2 \end{bmatrix}\] Out of the given matrix products, which one is not defined?

Let $\omega$ be the complex number $\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}$. 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 ___. (IIT-JEE, 2010)

Let A be a symmetric matrix such that $|A| = 2$ and $\begin{pmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{pmatrix} A = \begin{pmatrix} 1 & 2 \\ \alpha & \beta \end{pmatrix}$. If the sum of the diagonal elements of A is $s$, then $\frac{\beta s}{\alpha^2}$ is equal to ___.

If $x, y, z$ are in A.P., then the value of the determinant $\begin{vmatrix} a+2 & a+3 & a+2x \\ a+3 & a+4 & a+2y \\ a+4 & a+5 & a+2z \end{vmatrix}$ is

Value of \[\begin{vmatrix} 1+x_1 & 1+x_1 x & 1+x_1 x^2 \\ 1+x_2 & 1+x_2 x & 1+x_2 x^2 \\ 1+x_3 & 1+x_3 x & 1+x_3 x^2 \end{vmatrix}\] depends upon

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

If $ A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix} $, then $ \text{adj}(3A^2 + 12A) $ is equal to

If $A$ and $B$ are square matrices of order $n$, then $A - \lambda I$ and $B - \lambda I$ commute for every scalar $\lambda$, only if

Let the numbers 2, b and c be in an A.P. and $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & b & c \\ 4 & b^2 & c^2 \end{bmatrix}$. If $\det(A) \in [2, 16]$, then $c$ lies in the interval:

If $A = \frac{1}{2}\begin{pmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{pmatrix}$, then:

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

Let $A$ be a square matrix of order $3$ such that $A + A^T = \begin{bmatrix} 10 & 4 & 6 \\ 4 & 6 & -1 \\ 6 & -1 & 8 \end{bmatrix}$ where $a_{ij}, a_{ji}, a_{kj}$ are positive roots of the equation $x^3 - 6x^2 + px - 8 = 0, \forall c \in \mathbb{R}$, then the absolute value of $|A|$ is equal to

If determinant of matrix $ A $ is 5, then determinant of $2A$ is:

If determinant is symmetric, best method is:

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

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

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

Value of $\begin{vmatrix} x+y & z & z \\ x & y+z & x \\ y & y & z+x \end{vmatrix}$, where $x, y, z$ are nonzero real numbers, is equal to

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

If $\Delta(x) = \begin{vmatrix} \tan x & \tan(x+h) & \tan(x+2h) \\ \tan(x+2h) & \tan x & \tan(x+h) \\ \tan(x+h) & \tan(x+2h) & \tan x \end{vmatrix}$, then the value of $\lim_{h \to 0} \dfrac{\Delta(\pi/3)}{\sqrt{3}h^2}$ is

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

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

For Problems 14 and 15$A$ and $B$ are square matrices such that det.$(A) = 1$, $BB^T = I$, det.$(B) > 0$, and $A(\text{adj.}A + \text{adj.}B) = B$.The value of det.$(A + B)$ is

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

If A, B, C are nonsingular square matrices of order 3 × 3 then which of the following is not necessarily true?

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

The system of linear equations$x + y + z = 5$$x + 2y + 2z = 6$$x + 3y + \lambda z = \mu$has infinitely many solutions. Find $\lambda + \mu$.

If determinant is zero, matrix is:

If $A$ is an idempotent matrix satisfying $(I - 0.4A)^{-1} = I - \alpha A$, where $I$ is unit matrix of the same order as that of $A$, then the value of $\alpha$ is:

Choose the correct answer

Let $A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{pmatrix}(10)$ and $B = \begin{pmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{pmatrix}$. If $B$ is the inverse of matrix $A$, then $\alpha$ is

How many different diagonal matrices of order n can be formed which are involuntary?

If \[\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \cdots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix},\] then the inverse of $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is:

If nth-order square matrix A is an orthogonal, then $|\text{adj}(\text{adj } A)|$ is

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then $\det[(I+B)^{50} - 50B]$ is equal to

If determinant is triangular, then off-diagonal elements:

If $A$ is an invertible matrix, then $(\text{adj } A)^{-1}$ is equal to

For Problems 22–24Consider the system of equations$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = \mu$The system has no solution if

If $A = \begin{bmatrix} 1 & \tan x \\ -\tan x & 1 \end{bmatrix}$, then $A^T A^{-1}$ is

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and one row is replaced by sum of two rows, determinant becomes:

If $\det(A)$=5, then det(A\)^5A^3A⁻^7A^{-1}) equals:

If determinant has identical columns, then determinant equals:

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $(\alpha \in \mathbb{R})$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then a value of $\alpha$ is:

Determinants Questions (2072)