Determinants Questions (2072)

Let \( f(x) = \begin{vmatrix} \cos x & \sin x & \cos x \\ \cos 2x & \sin 2x & 2\cos 2x \\ \cos 3x & \sin 3x & 3\cos 3x \end{vmatrix} \). Then find the values of \( f'(0) \) and \( f'(\pi/2) \).

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

Given that matrix \(A = \begin{bmatrix} x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z \end{bmatrix}\). If \(xyz = 60\) and \(8x + 4y + 3z = 20\), then \(A(\text{adj }A)\) is equal to

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

Let A be a square matrix of order 3 such that \(\det(A) = \dfrac{1}{3}\), then the value of \(\det(\text{adj}\, A^{-1})\) is

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

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

For Problems 7 and 8Consider an arbitrary \(3 \times 3\) non-singular matrix \(A = [a_{ij}]\). A matrix \(B = [b_{ij}]\) is formed such that \(b_{ij}\) is the sum of all the elements except \(a_{ij}\) in the \(i\)th row of \(A\).If there exists a matrix \(X\) with constant elements such that \(AX = B\), then \(X\) is

For Problems 19–21Given that the system of equations \(x = cy + bz\), \(y = az + cx\), \(z = bx + ay\) has nonzero solutions and at least one of the \(a, b, c\) is a proper fraction.\(a^2 + b^2 + c^2\) is

Let \(f(x) = \begin{vmatrix} x^2 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}\) where \(p\) is a constant. Then \(\dfrac{d^3}{dx^3}[f(x)]\) at \(x = 0\) is

Let A be a matrix of order 3, such that ATA = I. Then find the value of det.(A2 − I).

Consider the following statementsStatement-1 : If A is an idempotent non-zero matrix and I is an identity matrix of the same order, such that (A + I)n = I + 127 A. (n ∈ N), then 'n' has 3 positive divisors.Statement-2 : Let A = 3x216x, B = [a b c] and C = (x+2)25x22x5x22x(x+2)22x(x+2)25x2 be three given matrices, where a, b, c and x ∈ R. Given that tr(AB) = tr(C) ∀ x ∈ R, where tr(A) denotes trace of A. Solving, we get a + b + c = 7.Then, which of the following options is/are correct ?

If A is a non-singular matrix and AT denotes the transpose of A, then :

If A and B are square matrices of order 3, then the true statement is/are (where I is unit matrix).(A) det (-A) = -det A(B) If AB is singular then atleast one of A or B is singular(C) det (A + I) = 1 + det A(D) det (2A) = 2^3 det A

If \[\begin{vmatrix} a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2 \end{vmatrix} = k(a-b)(b-c)(c-a),\] then find the value of \(k\).

If $A = \begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}$ and $B$ is the adjoint of $A$, then $\det(AB+2I)$ is (where $I$ is $3\times3$ identity)

Let α be a root of the equation x^2 + x + 1 = 0 and the matrix A = 1/sqrt(3) * [[1, 1, 1], [1, α, α^2], [1, α^2, α^4]], then the matrix A^31 is equal to:

The value of \(\sum_{r=2}^{n} (-2)^r \begin{vmatrix} ^{n-2}C_{r-2} & ^{n-2}C_{r-1} & ^{n-2}C_r \\ -3 & 1 & 1 \\ 2 & -1 & 0 \end{vmatrix}\) \((n > 2)\) is

The number of θ ∈ (0, 4π) for which the system of linear equations 3(sin 3θ)x - y + z = 2 3(cos 2θ)x + 4y + 3z = 3 6x + 7y + 7z = 9 has no solution is :

A is an involuntary matrix given by \(A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}\), then the inverse of \(A/2\) will be

The number of real values λ, such that the system of linear equations2x - 3y + 5z = 9x + 3y - z = -183x - y + (λ2 - |λ|)z = 16has no solution, is :-

The value of the determinant of a matrix is given by the expression. If the determinant is 1, what is the value?

Let A and B be 3 x 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A^2 B^2 - B^2 A^2)X = 0, where X is a 3 x 1 column matrix of unknown variables and 0 is a 3 x 1 null matrix, has :

If \((\omega \neq 1)\) is a cubic root of unity, then \[\begin{vmatrix} 1 & 1+i+\omega^2 & \omega^2 \\ 1-i & -1 & \omega^2-1 \\ -i & -1+\omega-i & -1 \end{vmatrix}\] equals

If A and B are two nonsingular matrices of the same order such that \(B^r = I\), for some positive integer \(r > 1\), then \(A^{-1} B^{r-1} A - A^{-1} B^{-1} A =\)

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

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 ____.

How many different diagonal matrices of order n can be formed which are idempotent?

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 :