Determinants Questions - JEE

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices : Tp = {A = abca : a, b, c ∈ {0, 1, 2, ..., p-1}}. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is -

If one root of the equation \[\begin{vmatrix} 7 & 6 & x^2-13 \\ 2 & x^2-13 & 2 \\ x^2-13 & 3 & 7 \end{vmatrix} = 0\] is $x = 2$, the sum of all other five roots is

Which of the following options is/are correct?(A) Let $A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ are two matrices such that $AB = (AB)^{-1}$ and $AB \neq I$ (where $I$ is an identity matrix of order $3 \times 3$). Then $tr\left(AB + (AB)^2 + (AB)^3 + ..... + (AB)^{100}\right)$ equals 100.(B) If $A$ and $B$ are square matrices of order 3, where $|A| = -2$ and $|B| = 1$, then $|(A^{-1})adj(B^{-1})adj(2A^{-1})|$ equals 8.(C) If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $F(x).F(y) = F(x + y)$ and $[F(x)]^{-1} = F(-x)$.(D) Let $X$ be the solution set of the equation $A^x = I$, where $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$ and $I$ is the corresponding unit matrix and $X \subseteq \mathbb{N}$ then the minimum value of $\sum(\cos^n \theta + \sin^n \theta), \theta \in \mathbb{R}, n \in \mathbb{Z}$ is 2.

Given the system of linear equations:$x - 2y + kz = 1$ (1)$2x + y + z = 2$ (2)$3x - y - kz = 3$ (3)For the system to have infinite solutions, the value of k is:

Let A, B, C, D be real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix M = ABCD, then M2016 is equal to

If $A$, $B$ and $C$ are the angles of a non-right angled triangle $ABC$, the value of \[\begin{vmatrix} \tan A & 1 & 1 \\ 1 & \tan B & 1 \\ 1 & 1 & \tan C \end{vmatrix}\] is

If the determinant <mfenced open="|

Let A be a 2 × 2 real matrix with entries from {0,1} and |A| ≠ 0. Consider the following two statements:(P) If A ≠ I2, then |A| = -1(Q) If |A| = 1, then tr(A) = 2,where I2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A.Then:

If the system of linear equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ has infinitely many solutions, then

If the system of equations $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to:

If A is a 3 × 3 matrix and u is a vector. If Au and u are orthogonal for all real u, then matrix A is a

Among the statements: I: If $\begin{vmatrix}1&\cos\alpha&\cos\beta\\\cos\alpha&1&\cos\gamma\\\cos\beta&\cos\gamma&1\end{vmatrix}=\begin{vmatrix}0&\cos\alpha&\cos\beta\\\cos\alpha&0&\cos\gamma\\\cos\beta&\cos\gamma&0\end{vmatrix}$, then $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\dfrac{3}{2}$ II: If $\begin{vmatrix}x^2+x&x+1&x-2\\2x^2+3x-1&3x&3x-3\\x^2+2x+3&2x-1&2x-1\end{vmatrix}=px+q$, then $p^2=196q^2$

Let $n$ be the number obtained on rolling a fair die. If the probability that the system $x-ny+z=6$ $x+(n-2)y+(n+1)z=8$ $(n-1)y+z=1$ has a unique solution is $\dfrac{k}{6}$, then the sum of $k$ and all possible values of $n$ is:

Let A = 02y12xy-12x-y1. If (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is :-

If the value of the determinant \[\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}\] is positive, then (for $a, b, c > 0$)

Let A be a symmetric matrix such that |A| = 2 and 2132A=12αβ. If the sum of the diagonal elements of A is s, then βsα2 is equal to ____.

Let $f(x) = \dfrac{1+x}{1-x}$. If A is a matrix for which $A^3 = O$, then $f(A)$ is

If the system of equation 2x + y - z = 5 2x - 5y + λz = μ x + 2y - 5z = 7 has infinitely many solutions, then (λ + μ)2 + (λ - μ)2 is equal to

For positive numbers x, y and z, the numerical value of the determinant 1logxylogxzlogyx1logyzlogzxlogzy1 is -

Let A be a 3 x 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 -> 2R2 + 5R3 on 2A, then det(B) is equal to :

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$, where $b_{ij} = 2^{i+j} a_{ij}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

If \(\ell will always be greater than -

Let A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} p \\ q \end{pmatrix} \neq \begin{pmatrix} 0 \\ 0 \end{pmatrix} are matrices satisfying AB = B and a + d = 5050. Find the value of (ad - bc).

255. Let $A = [a_{ij}]_{2 \times 2}$ be a matrix where $a_{ij} \in \{2, 3\}$. If determinant of matrix $A$ is non-negative, then probability that it is invertible is:

The greatest value of $c \in R$ for which the system of linear equations$x - cy - cz = 0$$cx - y + cz = 0$$cx + cy - z = 0$has a non-trivial solution, is:

Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and b3 respectively. If x1 = <mfenced open="[

If A and B are two 3 x 3 matrices such that their product AB is a null matrix then is/are always true (where XT denotes transpose of X and I denotes unit matrix)-(A) det. A ≠ 0 ⇒ B must be a null matrix.(B) det. B ≠ 0 ⇒ A must be a null matrix.(C) If none of A and B are null matrices then atleast one of the two matrices must be singular.(D) If neither det. A nor det. B is zero then the given statement is not possible.

If $\det(A)$=k, then det((A\)^0)^{-1}\cdotA) equals:

Let D_k = 12k2k-1nn2+n+2n2nn2+nn2+n+2. If ∑k=1nDk=96, then n is divisible by

Let p and p + 2 be prime numbers and let Δ = <mfenced open="|

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (A^T)^{20} is equal to

Which of the following is(are) NOT the square of a 3 × 3 matrix with real entries?(A) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (B) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (C) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (D) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let E = 12323481318, P = 100001010 and F = 132132243.If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE?

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then the trace of A is:

Let R = { a3bc2d050 : a, b, c, d ∈ {0, 3, 5, 7, 11, 13, 17, 19} }. Then the number of invertible matrices in R is

The matrix A2 + 4A - 5I, where I is an identity matrix and A = 124-3 equals :

The total number of distinct x ∈ R for which <mfenced open="|

Consider the matrices: A = \begin{pmatrix} 2 & -5 \\ 3 & m \end{pmatrix}, B = \begin{pmatrix} 20 \\ m \end{pmatrix} and X = \begin{pmatrix} x \\ y \end{pmatrix}. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 \int_{a}^{b} |A| dm is equal to ________.

The determinant x2y+z2yzy2z+x2zxz2x+y2xy is divisible by -

If A = 1000110-24, I = 100010001 and A-1 = 16(A2 + cA + dI), then the value of c + d is:

Which of the following options is/are correct ?(A) Let $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix}$ and $C_r = \begin{bmatrix} r.3^r & 2^r \\ 0 & (r-1)3^r \end{bmatrix}$ be 3 given matrices. Then $\sum_{r=1}^{50} tr.((AB)^r C_r) = 3(49.3^{50} + 1)$.(B) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $AX = A$, then $X = \begin{bmatrix} a & b \\ 2-2a & 1-2b \end{bmatrix}$ for $a, b \in R$(C) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XA = I$, then $X$ does not exist(D) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XB = 0$ but $BX \neq 0$, then $X = \begin{bmatrix} a & -3a \\ c & -3c \end{bmatrix}$, $a, c \in R, 3a + c \neq 0; 3b + d \neq 0$

If $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$, then the determinant of the matrix $\left(A^{2025}-3A^{2024}+A^{2023}\right)$ is

Evaluate \[\begin{vmatrix} {}^{x}C_1 & {}^{x}C_2 & {}^{x}C_3 \\ {}^{y}C_1 & {}^{y}C_2 & {}^{y}C_3 \\ {}^{z}C_1 & {}^{z}C_2 & {}^{z}C_3 \end{vmatrix}.\]

Let A = 100210321. If u1 and u2 are column matrices such that Au1 = 100 and Au2 = 010, then u1 + u2 is equal to:

Let $f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1+\sin 2x \end{vmatrix}, x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

28. $\sum_{r=1}^{10} D_r$ is

Let A be a square matrix such that $A(\text{adj. }A) = \begin{bmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 4\end{bmatrix}$. Find the value of $|\text{adj. }(\text{adj. }A)|$.

If P is a 3 x 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 x 3 identity matrix, then there exists a column matrix X = xyz ≠ 000 such that

For a real number $\alpha$, if the system $$\begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$$ of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 = $

Determinants Questions (2072)