Determinants Questions - JEE

If $x, y, z$ distinct common roots of $z^6 - 1 = 0$ and $z^3 - 1 = 0$ then $$\begin{vmatrix} x - y - z & 2x & 2x \\ 2y & y - x - z & 2y \\ 2z & 2z & z - x - y \end{vmatrix}$$ is equal to _____.

If a2 + b2 + c2 = -2 and f(x) = 1+a2x(1+b2)x(1+c2)x(1+a2)x1+b2x(1+c2)x(1+a2)x(1+b2)x1+c2x then f(x) is a polynomial of degree-

The value of k for which the set of equations 3x + ky - 2z = 0, x + ky + 3z = 0 and 2x + 3y - 4z = 0 has a non-trivial solution is-

Let a, b, c be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x = cy + bz, y = az + cx and z = bx + ay, then a^2 + b^2 + c^2 + 2abc is equal to

The system of equations$x + 8y + 7z = 0$$9x + 2y + 3z = 0$$x + y + z = 0$has a non-trivial solution. If $x = a, y = b, z = c$ is the solution and $(a, b, c)$ lies on the plane $x + 2y + z = 6$, find the value of $2a + b + c$.

The determinant <mfenced open="|

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

System of equation x + y + az = b, 2x + 3y = 2a & 3x + 4y + a^2z = ab + 2 has

If the system of equations$(a-t)x + by + cz = 0$$bx + (c-t)y + az = 0$$cx + ay + (b-t)z = 0$has non-trivial solution, then product of all possible values of $t$ is

System of linear equations in x, y, z have infinite solutions which2x + y + z = 1x - 2y + z = 23x - y + 2z = 3(A) can be written as (-3λ -1, λ, 5λ + 3) ∀ λ ∈ R(B) can be written as (3λ -1, -λ, -5λ + 3) ∀ λ ∈ R(C) are such that every solution satisfy x - 3y + 1 = 0(D) are such that none of them satisfy 5x + 3z = 1

Let a, b, c are the solutions of the cubic x3 - 5x2 + 3x - 1 = 0, then find the value of the determinant

Consider the system of linear equation x + y + z = 4μ, x + 2y + 2λz = 10μ, x + 3y + 4λ²z = μ² + 15 where λ, μ ∈ R. Which one of the following statements is NOT correct?

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

The set of all values of λ for which the system of linear equations :2x1 - 2x2 + x3 = λx1, 2x1 - 3x2 + 2x3 = λx2, -x1 + 2x2 = λx3has a non-trivial solution

Let the system of linear equations 4x + λy + 2z = 0 2x - y + z = 0 μx + 2y + 3z = 0, λ, μ ∈ R has a non-trivial solution. Then which of the following is true ?

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |A^T adj(adj(2A))^-1 (adj(4B))(adj(AB))^-1 A^T| is equal to :

921. If $A$ and $B$ are square matrices of order 3 such that $2(A + B) = A^T + B^T + 3I$ and $AA^T = 4I$, then find the value of $\det(12A^{-1} - BA^T + I)$.[Note: $I$ is an identity matrix of order 3 and $P^T$ denotes the transpose of matrix $P$.]

Let $P=\begin{bmatrix}0&2&\lambda\\2&3&1\\1&\mu&3\end{bmatrix}$ and $\text{Adj}(P)=\begin{bmatrix}10&-7&-1\\-5&-1&2\\-5&2&-4\end{bmatrix}$. Then $\left|(\text{adj}P)^{-1}+14\,\text{adj}(P^{-1})\right|$ equals

Roots of the equation $\begin{vmatrix} x & m & n & 1 \\ a & x & n & 1 \\ a & b & x & 1 \\ a & b & c & 1 \end{vmatrix} = 0$ are

Let $A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}$. If $u_1$ and $u_2$ are column matrices such that $Au_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$ and $Au_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, then $u_1 + u_2$ is equal to:

If A, B and C are n x n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the det(A2 BC-1) is equal to

The matrix 2-2-4-1341-2-3 is (A) non-singular (B) Idempotent (C) Nilpotent (D) Involutory

If an idempotent matrix is also skew symmetric then it must be-

If \[\begin{vmatrix} a^2 & b^2 & c^2 \\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{vmatrix} = k\lambda \begin{vmatrix} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}\]then $k$ equals

$$\begin{vmatrix} ^5C_1 & ^5C_2 & ^5C_3 \\ ^6C_1 & ^4C_2 & ^4C_3 \\ ^3C_1 & ^3C_2 & ^3C_3 \end{vmatrix} \begin{vmatrix} ^5C_1 & ^6C_2 & ^7C_3 \\ ^4C_1 & ^5C_2 & ^6C_3 \\ ^1C_1 & ^4C_2 & ^5C_3 \end{vmatrix} = $$ _____.

AB = A and BA = B, then (here A & B are matrix of n x n) which of the following must be true -

Let $A = [a_{ij}]$, $a_{ij} \in \mathbb{Z} \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices A such that the sum of all entries is a prime number $p \in (2,13)$ is ___.

The system of linear equations$x + y + z = 6$$4x + \lambda y - \lambda z = \lambda - 2$$3x + 2y - 4z = -5$has a solution. Then $\lambda$ equals

If $\begin{vmatrix}x-4&2x&2x\\2x&x-4&2x\\2x&2x&x-4\end{vmatrix}=(A+Bx)(x-A)^2$, then the ordered pair $(A,B)$ is equal to

If D1 and D2 are two 3 × 3 diagonal matrices where none of the diagonal element is zero, then -

Let det(adj(adjA)) = 14^4 where A = x2-1-1122-11, x ≠ -25/3, then

Consider the following statements.Statement-1 : If $\begin{bmatrix} 3 & -2 \\ 3 & 0 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} y & y \\ x & x \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ 3y & 3y \\ 10 & 10 \end{bmatrix}$, then $2x + 3y = \lambda$.Statement-2 : Given that $\ell + 5 = p + 2m$, where $A$ is a square matrix of order $n$.$\ell$ = maximum number of distinct entries if $A$ is a triangular matrix.$m$ = maximum number of distinct entries if $A$ is a diagonal matrix.$p$ = minimum number of zeroes if $A$ is a triangular matrix.Statement-3 : Let $A$ be the set of all $3 \times 3$ skew symmetric matrices whose entries are either $-1, 0$ or $1$. If there are exactly three $0$'s, three $1$'s and three $(-1)$'s, then number of such matrices is equal to $\mu$.Then, which of the following options is/are correct?

The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0 is

Let $\Delta_1 = \begin{vmatrix} a & b & c \\ e & d & c+d \\ a & b & a+b+c \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} a & b & a+c \\ b & d & b+d \\ a & c & a+d+c \end{vmatrix}$ then the value of $\left|\frac{\Delta_1}{\Delta_2}\right|$, where $b \neq 0$ and $ad \neq bc$ is ____

If $f(x) = \begin{vmatrix} \cos x & x & 1 \\ 2\sin x & x^2 & 2x \\ \tan x & x & 1 \end{vmatrix}$, then $\lim_{x \to 0} \dfrac{f'(x)}{x}$

Let $$\begin{vmatrix} x^2 + 3x & x - 1 & x + 3 \\ x + 1 & -2x & x - 4 \\ x - 3 & x + 4 & 3x \end{vmatrix} = ax^4 + bx^3 + cx^2 + dx + e$$ be an identity in $x$, then $-\left[\frac{a + b + c + d + e}{a + e}\right]$ is _____ . (where $[.]$ denotes greatest integer function).

The number of A in Tp such that det (A) is not divisible by p is -

If determinant has a triangular form, its value is:

If the system of linear equations x - 4y + 7z = g 3y - 5z = h -2x + 5y - 9z = k is consistent, then :

The value of the determinant of a 3x3 matrix is given by the expression. If the matrix is singular, what is the value of the determinant?

If $A$, $B$, $A+I$, $A+B$ are idempotent matrices, then $AB$ is equal to

An invertible matrix A of order 3 satisfies the relation A = A-1 + 2I, (where I denotes identity matrix). The value of |A - I|.|A + I|.|A - 2I| is

Let $ A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix} $. If $ u_1 $ and $ u_2 $ are column matrices such that $ Au_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} $ and $ Au_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} $, then $ u_1 + u_2 $ is equal to

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

Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^4 = aA + bI, then the value of a + b 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="[

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 :

Evaluate \[\begin{vmatrix} \cos\alpha\cos\beta & \cos\alpha\sin\beta & -\sin\alpha \\ -\sin\beta & \cos\beta & 0 \\ \sin\alpha\cos\beta & \sin\alpha\sin\beta & \cos\alpha \end{vmatrix}\]

If $A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$, $A^{-1} = \alpha A + \beta I$ and $\alpha + \beta = -2$, then $4\alpha^2 + \beta^2 + \lambda^2$ is equal to:

Let a and b be two real numbers such that $a > 1$, $b > 1$. If $A = \begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$, then $\lim_{n \to \infty} (A^n)^{-1}$ is

Determinants Questions (2072)