Determinants Questions - JEE

Consider the system of linear equations:$x_1 + 2x_2 + x_3 = 3$$2x_1 + 3x_2 + x_3 = 3$$3x_1 + 5x_2 + 2x_3 = 1$The system has

If A is a nonsingular matrix such that $AA^T = A^T A$ and $B = A^{-1} A^T$, then matrix B is

If [ ] denotes the greatest integer less than or equal to the real number under consideration, and \(-1 \le x

For which of the following matrices, the number of left inverses is greater than the number of right inverses?

In triangle $ABC$, if $\begin{vmatrix} 1 & 1 & 1 \\ \cot\dfrac{A}{2} & \cot\dfrac{B}{2} & \cot\dfrac{C}{2} \\ \tan\dfrac{B}{2}+\tan\dfrac{C}{2} & \tan\dfrac{C}{2}+\tan\dfrac{A}{2} & \tan\dfrac{A}{2}+\tan\dfrac{B}{2} \end{vmatrix} = 0$, then the triangle must be

We have \[\Delta = \begin{vmatrix} \log x & \log y & \log z \\ \log p & \log q & \log r \\ \log l & \log m & \log n \end{vmatrix}\] Using the column operations $C_2 \to C_2 - C_1$ and $C_3 \to C_3 - C_2$, find the value of $\Delta$.

The system of linear equations$x + y + z = 2$$2x + 3y + 2z = 5$$2x + 3y + (a^2 - 1)z = a + 1$has a solution. Find the condition on $a$.

For Problems 9–11Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ satisfies $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$. And trace of a square matrix $X$ is equal to the sum of elements in its principal diagonal.Further consider a matrix $U_{3\times 3}$ with its columns as $U_1, U_2, U_3$ such that\[A^{50}U_1 = \begin{bmatrix}1\\25\\25\end{bmatrix},\quad A^{50}U_2 = \begin{bmatrix}0\\1\\0\end{bmatrix},\quad A^{50}U_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}\]The value of $|A^{50}|$ equals

Let $A = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}$. If for some $\theta \in (0, \pi)$, $A^2 = A^T$, then the sum of the diagonal elements of the matrix $(A + I)^3 + (A - I)^3 - 6A$ is equal to _____.

If $\begin{vmatrix} a & a^2 & 1+a^3 \\ b & b^2 & 1+b^3 \\ c & c^2 & 1+c^3 \end{vmatrix} = 0$ and $(1, a, a^2)$, $(1, b, b^2)$, $(1, c, c^2)$ are non-coplanar, then $abc$ equals:

If $A = \begin{bmatrix} 1 & \sin\theta & 1 \\ -\sin\theta & 1 & \sin\theta \\ -1 & -\sin\theta & 1 \end{bmatrix}$; then for all $\theta \in \left(\dfrac{3\pi}{4}, \dfrac{5\pi}{4}\right)$, $\det(A)$ lies in the interval:

If \[\begin{vmatrix} 6i & -3i & 1 \\ 4 & 3i & -1 \\ 20 & 3 & i \end{vmatrix} = x + iy\] then

In which of the following type of matrix inverse does not exist always?

If $Z$ is an idempotent matrix, then $(I + Z)^n$

If \[A = \begin{bmatrix} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{bmatrix},\] then $A$ is:

If matrix $A = [a_{ij}]_{3 \times 3}$, matrix $B = [b_{ij}]_{3 \times 3}$, where $a_{ij} + a_{ji} = 0$ and $b_{ij} - b_{ji} = 0\ \forall\ i, j$, then $A^4 B^3$ is:

Let A and B be symmetric matrices of same order. Then AB – BA is

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A \cdot \text{adj}\, A = AA^T$, then $5a + b$ is equal to

If $a, b, c$ are in G.P. with common ratio $r_1$ and $\alpha, \beta, \gamma$ are in G.P. with common ratio $r_2$, and equations $ax + \alpha y + z = 0$, $bx + \beta y + z = 0$, $cx + \gamma y + z = 0$ have only zero solution, then which of the following is not true?

If $\Delta = \begin{vmatrix} 1 & 1+i+\omega^2 & \omega^2 \\ 1-i & -1 & \omega^2-1 \\ -i & -1+\omega-i & -1 \end{vmatrix}$, then $\Delta$ equals:

Find the value of \lambda such that the system of equations has infinitely many (non-trivial) solutions:3x - 2y + z = 0\lambda x - 14y + 15z = 0x + 2y - 3z = 0

If $a, b, c$ are the roots of the equation $x^3 + 2x^2 + 1 = 0$, then $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} =$

Given \[\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 2\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 3\\0 & 1\end{bmatrix}\cdots\begin{bmatrix}1 & n-1\\0 & 1\end{bmatrix} = \begin{bmatrix}1 & 78\\0 & 1\end{bmatrix}\] Then find the inverse of matrix $A = \begin{bmatrix}1 & 13\\0 & 1\end{bmatrix}$. What is $A^{-1}$?

If $\begin{vmatrix} x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3 \end{vmatrix} = \begin{vmatrix} 2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x \end{vmatrix} = \begin{vmatrix} 4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5 \end{vmatrix} = 0$, then $x$ is equal to

If a determinant of order $3 \times 3$ is formed by using the numbers 1 or −1, then the minimum value of the determinant is

A and B are different matrices of order n satisfying A3 = B3 and A2B = B2A. If det.(A − B) ≠ 0, then find the value of det.(A2 + B2).

Elements of a matrix A of order $10 \times 10$ are defined as $a_{ij} = \omega^{i+j}$ (where $\omega$ is imaginary cube root of unity), then trace (A) of the matrix is

If $a^2 + b^2 + c^2 = -2$ and $f(x) = \begin{vmatrix} 1+a^2x & (1+b^2)x & (1+c^2)x \\ (1+a^2)x & 1+b^2x & (1+c^2)x \\ (1+a^2)x & (1+b^2)x & 1+c^2x \end{vmatrix}$, then $f(x)$ is a polynomial of degree

Let $ P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix} $ and $ Q = [q_{ij}] $ be two $ 3 \times 3 $ matrices such that $ Q - P^5 = I_3 $. Then $ \dfrac{q_{21} + q_{31}}{q_{32}} $ is equal to:

If $D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1+x & 1 \\ 1 & 1 & 1+y \end{vmatrix}$, then $D$ is divisible by:

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be such that $A^2 = I$, where $I$ is the identity matrix. Consider the following statements:Statement-1: $\det(A) = -1$Statement-2: $\text{tr}(A) = 0$Which of the following is correct?

The number of values of k for which the system of linear equations,$(k+2)x+10y=k$$kx+(k+3)y=k-1$has no solution, is

$\begin{vmatrix} ^xC_r & ^xC_{r+1} & ^xC_{r+2} \\ ^yC_r & ^yC_{r+1} & ^yC_{r+2} \\ ^zC_r & ^zC_{r+1} & ^zC_{r+2} \end{vmatrix}$ is equal to

Find the value of $\lambda$, if the following equations are consistent: $x + y - 3 = 0$; $(1 + \lambda)x + (2 + \lambda)y - 8 = 0$; $x - (1 + \lambda)y + (2 + \lambda) = 0$

If $\begin{vmatrix} b+c & c+a & a+b \\ a+b & b+c & c+a \\ c+a & a+b & b+c \end{vmatrix} = k \begin{vmatrix} a & b & c \\ c & a & b \\ b & c & a \end{vmatrix}$, then the value of $k$ is

For Problems 10–12\[f(x) = \begin{vmatrix} x+c_1 & x+a & x+a \\ x+b & x+c_2 & x+a \\ x+b & x+b & x+c_3 \end{vmatrix}\] and $g(x) = (c_1 - x)(c_2 - x)(c_3 - x)$Which of the following is not a constant term in $f(x)$?

Let $A$ and $B$ be two symmetric matrices of order 3.Statement-1: $A(BA)$ and $(AB)A$ are symmetric matrices.Statement-2: $AB$ is symmetric matrix if matrix multiplication of $A$ and $B$ is commutative.

Find the number of all possible matrices of order 3 × 3 with each entry 0 or 1. How many of these are symmetric?

Let \(x

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

The number of solutions of the set of equations \frac{2x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0, \frac{x^2}{a^2} + \frac{2y^2}{b^2} - \frac{z^2}{c^2} = 0, \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{2z^2}{c^2} = 0 is

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

If f(x) = \begin{vmatrix} (1-x)a & (1-2x)b & 1 \\ 1 & (1-x)a & (1-2x)b \\ b & a & (1-2x) \end{vmatrix}, where a, b are positive integers, then:

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} 2 & 1 \\ -4 & -2 \end{pmatrix}, then I + 2A + 3A^2 + \cdots + \infty is

If A = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix}, then A 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 -

Let $A_k=[a_{ij}]$ be square matrix of order 3 with $a_{ij}=(i-j)^k$ for all $i,j\in\{1,2,3\}$. Determinant value of $|A_1+A_3+A_5+\cdots+A_{2023}|$ equals

If a, b and c are sides of $\triangle ABC$ such that\[a^3 + b^3\cos B + c^3\cos A + b^2 + c^2 = 0\]\[c^3 + a^3\cos B + a^2\cos A + c^2 + a^2 = 0\]\[b^3 + a^3\cos B + b^3\cos A + a^2 + b^2 = 0\]where $\phi, \psi, \omega \in \mathbb{R}$ and $\angle A, \angle B, \angle C \in [\frac{\pi}{8}, \frac{7\pi}{8}]$, then $\triangle ABC$ is

Let A = \begin{pmatrix} m & n \\ p & q \end{pmatrix}, d = |A| \neq 0, |A - d(\text{adj } A)| = 0. Then

Determinants Questions (2072)