Matrices & Determinants Questions - JEE

The number of positive integral solutions of the equation $\begin{vmatrix} x^3+1 & x^2y & x^2z \\ xy^2 & y^3+1 & y^2z \\ xz^2 & yz^2 & z^3+1 \end{vmatrix} = 11$ is

$A$ and $B$ are square matrices of order $n$ such that $A^2 - B^2 = (A-B)(A+B)$. Which of the following must be true?

If a1, a2, a3, 5, 4, a6, a7, a8, a9 are in H.P., and \[D = \begin{vmatrix} a_1 & a_2 & a_3 \\ 5 & 4 & a_6 \\ a_7 & a_8 & a_9 \end{vmatrix}\] then the value of $[D]$ is (where $[\cdot]$ represents the greatest integer function) ________.

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by tr$(A)$, the sum of diagonal entries of $A$. Assume that $A^2 = I$.Statement-1: If $A \neq I$ and $A \neq -I$, then $\det A = -1$.Statement-2: If $A \neq I$ and $A \neq -I$, then $\text{tr}(A) \neq 0$.

The number of right inverses for the matrix $\begin{bmatrix} 1 & -1 & 2 \\ 2 & -1 & 1 \end{bmatrix}$ is

Let $A$ be a $3 \times 3$ matrix such that $A\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$. Then $A^{-1}$ 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:

Given \[\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix} = 0\] Find the sum of real roots.

If $A = \begin{bmatrix}0 & \tan\alpha/2\\ -\tan\alpha/2 & 0\end{bmatrix}$ and $I$ is a $2 \times 2$ unit matrix, then $(I - A)\begin{bmatrix}\cos\alpha & -\sin\alpha\\ \sin\alpha & \sin\alpha\end{bmatrix}$ is

Let P and Q be 3 × 3 matrices $P \neq Q$. If $P^3 = Q^3$ and $P^2Q = Q^2P$, then determinant of $(P^2 + Q^2)$ is equal to:

Let $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 5$ and $\Delta' = \begin{vmatrix} b_2c_3 - b_3c_2 & a_2c_3 - a_3c_2 & a_2b_3 - a_3b_2 \\ b_3c_1 - b_1c_3 & a_3c_1 - a_1c_3 & a_3b_1 - a_1b_3 \\ b_1c_2 - b_2c_1 & a_1c_2 - a_2c_1 & a_1b_2 - a_2b_1 \end{vmatrix}$. Find $\Delta'$.

If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then which one of the following holds for all $n \geq 1$, by the principle of mathematical induction?

If $K = 4\Delta^3$, where $\Delta$ is the determinant of a $3 \times 3$ matrix, find $K$.

Find $K = 4\Delta^3$

Given $A = \begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$and $A^{32} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$, then $\alpha =$

If A and B are square matrices of the same order and A is nonsingular, then for a positive integer n, $(A^{-1} B A)^n$ is equal to

Consider the system of equations: $\lambda x + y + z = 1$; $x + \lambda y + z = \lambda$; $x + y + \lambda z = \lambda^2$.Now, match the following lists:List Ia. $\lambda = 1$b. $\lambda \neq 1$c. $\lambda \neq 1, \lambda \neq -2$d. $\lambda = -2$List IIp. unique solutionq. infinite solutionr. No solutionCodes:(1) a-q, b-p, c-r, d-r(2) a-r, b-p, c-q, d-r(3) a-r, b-r, c-q, d-p(4) a-q, b-p,r, c-p, d-r

The number of distinct real roots of $\begin{vmatrix} \sin x & \cos x & \cos x \\ \cos x & \sin x & \cos x \\ \cos x & \cos x & \sin x \end{vmatrix} = 0$ in the interval $-\pi/4 \leq x \leq \pi/4$ is

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

Matrices & Determinants Questions (2045)