Determinants Questions (2072)

Let the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ satisfy $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ is:

If in the determinant Δ = a1b1c1a2b2c2a3b3c3, A1, B1, C1 etc. be the co-factors of a1, b1, c1 etc., then which of the following relations is incorrect-

If matrix \(A\) is given by \(A = \begin{bmatrix} 6 & 11 \\ 2 & 4 \end{bmatrix}\), then the determinant of \(A^{2005} - 6A^{2004}\) is

Let \(P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}\), \(Q = PAP^T\) and \(X = P^T Q^{2005} P\). Find \(X\), given \(A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\).

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

If \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\) (where \(bc \neq 0\)) satisfies the equations \(x^2 + k = 0\), then

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

The system of equations:\(\alpha x + y + z = \alpha - 1\)\(x + \alpha y + z = \alpha - 1\)\(x + y + \alpha z = \alpha - 1\)has no solution, if \(\alpha\) is

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:

If A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}, then A^2 - 4A - 5I is equal to

If \[\Delta_r = \begin{vmatrix} r & 612 & 915 \\ 101r^2 & 2r & 3r \\ r & \dfrac{1}{r} & \dfrac{1}{r^2} \end{vmatrix}\], then the value of \(\displaystyle\lim_{n \to \infty} \dfrac{1}{n^3} \sum_{r=1}^{n} \Delta_r\) is ________.

If \(y = \sin mx\), then the value of the determinant \[\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix}\] where \(y_n = \dfrac{d^n y}{dx^n}\) is

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 $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & -2 & 3 \end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then if $A^7-4A^6+6A^5=\alpha A^2+\beta A+\gamma I$ then $(\alpha, \beta, \gamma)$ is:

For Problems 16–18Consider the polynomial function\[f(x) = \begin{vmatrix} (1+x)^a & (1+2x)^b & 1 \\ 1 & (1+x)^a & (1+2x)^b \\ (1+2x)^b & 1 & (1+x)^a \end{vmatrix}\]\(a, b\) being positive integers.The coefficient of \(x\) in \(f(x)\) is

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

If \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\), then \(A =\)

If $$\begin{vmatrix} x & x+y & x+y+z \\ 2x & 3x+2y & 4x+3y+2z \\ 3x & 6x+3y & 10x+6y+3z \end{vmatrix} = -64$$, then the real value of $x$ is ____

If P is a $3 \times 3$ real matrix such that $P^T = aP + (a-1)I$, where $a > 1$, then

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

The values of α, for which 13/2α+3/211/3α+1/32α+33α+10 = 0, lie in the interval

If a, b, c are sides of a scalene triangle, then the value of abcbcacab is :

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