Determinants Questions - JEE

If determinant is zero but matrix not zero matrix, rank is:

The determinant $\begin{vmatrix} y^2 & -xy & x^2 \\ a & b & c \\ a' & b' & c' \end{vmatrix}$ is equal to

The number of values of k for which the linear equations$4x + ky + 2z = 0$$kx + 4y + z = 0$$2x + 2y + z = 0$possess a non-zero solution is

Let A be a 3x3 matrix and det(A) = 2. If n = det(adj(adj(.....(adj(A))))), where adj is applied 2024 times, then the remainder when n is divided by 9 is equal to ________.

If determinant is zero, then system of linear equations is:

In a $\triangle ABC$, if $$\begin{vmatrix} 1 & a & b \\ 1 & c & a \\ 1 & b & c \end{vmatrix} = 0$$, then $\sin^2 A + \sin^2 B + \sin^2 C = $_____.

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

Let $M = \begin{bmatrix} 0 & 1 & a \\ 1 & 2 & 3 \\ 3 & b & 1 \end{bmatrix}$ and $adjM = \begin{bmatrix} -1 & 1 & -1 \\ 8 & -6 & 2 \\ -5 & 3 & -1 \end{bmatrix}$ where $a$ and $b$ are real numbers. Which of the following options is/are correct?(A) $a + b = 3$(B) $\det(adjM^2) = 81$(C) $(adjM)^{-1} + adjM^{-1} = -M$(D) If $M \begin{bmatrix} \alpha \\ 1 \\ \gamma \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$, then $\alpha - \beta + \gamma = 3$

If $\det(A)$=5, then det(A\)^2A^2A⁻^3) equals:

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

For Problems 1–3Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = O$.$\text{tr}(A)$ is equal to

Let β be a real number. Consider the matrixA = β0121-231-2If A7 - (β-1)A6 - βA5 is a singular matrix, then the value of 9β is ____.

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

If $A$ and $B$ are square matrices of order 3 such that $\det(A) = -2$ and $\det(B) = 1$, then find the value of $\det\left(A^{-1} \cdot \text{adj}(B^{-1}) \cdot \text{adj}(2A^{-1})\right)$.

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

If $ A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix} $ and $ B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix} $, then the value of $ \alpha $ for which $ A^2 = B $ is

The value of the determinant of a matrix is given by the answer key.

Let A = [aij] be a 3x3 matrix such that aij = 2i-j for all i, j. Then the matrix An for any positive integer n is equal to:

Let A be a 3x3 matrix such that A^2 - 5A + 7I = 0. If A^n = 5^n A - 7^n I for some n, then n is equal to:

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 $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ & $P = \begin{bmatrix} \cos \frac{\pi}{12} & \sin \frac{\pi}{12} \\ -\sin \frac{\pi}{12} & \cos \frac{\pi}{12} \end{bmatrix}$ and $Q = P^T AP$, then if $PQ^{2014}P^T = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ then sum of digits of $b$ is _____.

Let A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}. If M and N are two matrices given by M = \sum_{k=1}^{10} A^{2k} and N = \sum_{k=1}^{10} A^{2k-1} then MN^2 is (1) a non-identity symmetric matrix (2) a skew-symmetric matrix (3) neither symmetric nor skew-symmetric (4) an identity matrix

If $A = \begin{bmatrix}i & -i\\ -i & i\end{bmatrix}$ and $B = \begin{bmatrix}1 & -1\\ -1 & 1\end{bmatrix}$, then $A^8$ equals

The value of an odd order determinant in which aij + aji = 0 ∀ i, j is -

If $S_r = \begin{vmatrix} 2r & x & n(n+1) \\ 6r^2-1 & y & n^2(2n+3) \\ 4r^3-2nr & z & n^3(n+1) \end{vmatrix}$, then $\sum_{r=1}^n S_r$ does not depend on -

If a, b, c are sides of a scalene triangle, then the value of <mfenced close="|

If S is the set of distinct values of b for which the following system of linear equations has no solution, then S is\[x + y + z = 1\]\[x + ay + z = 1\]\[ax + by + z = 0\]

The determinant <mfenced close="|

The system of linear equations$x + \lambda y - z = 0$$\lambda x - y - z = 0$$x + y - \lambda z = 0$has a non-trivial solution for

Find the number of A = [aij]2x2 satisfying aij is 1 or -1 and a11a21 + a12a22 = 0.

The number of values of x for which the matrix $A = \begin{pmatrix} 3-x & 2 & 2 \\ 2 & 4-x & 1 \\ -2 & -4 & -1-x \end{pmatrix}$ is singular, 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

If $\mathbf{A} = \begin{bmatrix} ab & b^2 \\ -a^2 & -ab \end{bmatrix}$, then A is a/an

If the system of equations x + y - 3 = 0, (1 + K)x + (2 + K)y - 8 = 0 & x - (1 + K)y + (2 + K) = 0 is consistent then the value of K may be -

If determinant is diagonal matrix, its value is:

$a, b, c$ are distinct real numbers, not equal to one. If $ax + y + z = 0$, $x + by + z = 0$, and $x + y + cz = 0$ have a non-trivial solution, then the value of $\dfrac{1}{1-a} + \dfrac{1}{1-b} + \dfrac{1}{1-c}$ is equal to

If the trivial solution is the only solution of the system of equations$x + ky + z = 0$$kx + 3y + kz = 0$$3x + y + z = 0$Then, the set of values of $k$ is

Let $M$ denote the matrix $\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$, where $i^2 = -1$, and let $I$ denote the identity matrix $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Then the matrix $I + M + M^2 + M^3 + M^4 + \ldots + M^{2010}$ is equal to:

Find the sum of all positive integral values of a for which every solution to the system of equation x + ay = 3 and ax + 4y = 6 satisfy the inequalities x > 1, y > 0.

Let f(x)=1x-1x2(x-1)(x-1)(x-2)x(x-1)3(x-1)(x-2)(x-1)(x-2)(x-3)x(x-1)(x-2) & Dr=1102r70173r+1111. The value of ∑r=110Dr is

Let a1, a2, a3, ..., a10 be in G.P. with ai > 0 for i = 1,2,..., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for which . Then the number of elements in S, is :

For each real number x such that -1 A(x) = ⎡⎣1 -x⎤⎦ and let y and z = x/(1-x) and 1/(1-x). Then

Let us consider the following matrix: \[\begin{bmatrix} 1 & X & X \\ X & 1 & X \\ X & X & 1 \end{bmatrix}\] which are six non-singular matrices because six blanks (i.e., $X$) shall be filled by five zeros and one 1. In the same manner, we have the matrix: \[\begin{bmatrix} X & X & 1 \\ X & 1 & X \\ 1 & X & X \end{bmatrix}\] which are six non-singular matrices. Therefore, in the required case, there are more than 7. How many such matrices are possible?

Let A = [aij] be a 3x3 matrix such that AT = A and det(A) = 0. If the sum of the diagonal elements of A is 6 and the sum of the squares of the diagonal elements is 14, then the possible value(s) of det(A + I) is/are

The number of values of x in the closed interval [−4, −1], for which the matrix $\begin{pmatrix} 3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2 \end{pmatrix}$ is singular, is

If $f(x) = \begin{vmatrix} 3 & 3x & 3x^2 + 2a^2 \\ 3x & 3x^2 + 2a^2 & 3x^3 + 6a^2x \\ 3x^2 + 2a^3 & 3x^3 + 6a^2x & 3x^4 + 12a^2x^2 + 2a^4 \end{vmatrix}$, then which is true?

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

Let $A = \begin{pmatrix}1&1\\1&1\end{pmatrix}$. If $\det(A^n - I) = 1 - \lambda^n$, find $\lambda$.

Question 83: Consider the determinant $\Delta = \begin{vmatrix} a_1 + b_1 x^2 & a_1 x^2 + b_1 & c_1 \\ a_2 + b_2 x^2 & a_2 x^2 + b_2 & c_2 \\ a_3 + b_3 x^2 & a_3 x^2 + b_3 & c_3 \end{vmatrix} = 0$, where $a_i, b_i, c_i \in \mathbb{R}$ and $x \in \mathbb{R}$.

The value of the determinant of a matrix is given by the answer key.

Determinants Questions (2072)