Determinants Questions (2072)

Let \( f(t) = \begin{vmatrix} \cos t & t & 1 \\ 2\sin t & t & 2t \\ \sin t & t & t \end{vmatrix} \). Then find \( \lim_{t \to 0} \dfrac{f(t)}{t^2} \).

If \(\Delta = \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix}\), then \(\Delta\) equals:

The value of determinant \(\begin{vmatrix} (a^x+a^{-x})^2 & (a^x-a^{-x})^2 & 1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}\) is

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

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then find the value of det(A - I).

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is (are) TRUE?

Let S be the set of all column matrices \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} such that b_1, b_2, b_3 \in \mathbb{R} and the system of equations (in real variables) -x + 2y + 5z = b_1 2x - 4y + 3z = b_2 x - 2y + 2z = b_3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \in S?

If A = ete-t cos te-t sin tet-e-t cos t - e-t sin t-e-t sin t + e-t cos tet2e-t sin t-2e-t cos t Then A is -

If $M = \begin{pmatrix} 2 & ABC^{2020} \\ D \end{pmatrix}$ (Given), $|M| = |2ABC^{2020}D|$, and $ABC^{2020}$ is a $2 \times 2$ matrix, find $|2ABC^{2020}D|$ if $|ABC^{2020}D| = 4096$.

If $A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$ and $(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, then $13(a^2 + b^2)$ is equal to ____.

Find the adjugate matrix determinant when $|M| = 2 \times$ Area of triangle with vertices $(a,d)$, $(b,e)$, $(c,f)$ with sides 6, 8, 10.

If $|A \text{ }adj B| \text{ }adj| (3A^{-1})|$, find the determinant.

Let S = {A = 01c1ad1be : a, b, c, d, e ∈ {0, 1} and |A| ∈ {-1, 1}}, where |A| denotes the determinant of A. Then the number of elements in S is ____.

If system of equation a1x + b1y = c1 & a2x + b2y = c2 (where a1, b1, c1, a2, b2, c2 ≠ 0) has infinite solutions, then-(A) a1/a2 = b1/b2 = c1/c2(B) a1 + a2/a1 - a2 = b1 + b2/b1 - b2 = c1 + c2/c1 - c2(C) the quadratic equations a1x2 + b1x + c1 = 0 & a2x2 + b2x + c2 = 0 have no common root(D) system of equation a12 a2x + b12 b2y = c12 c2 & a1 a22 x + b1 b22 y = c1 c22 will also have infinite number of solutions

If det(A)=2, then det(\(A\)^T\)\(A\)^{-1}) equals:

Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible, if

Given: \(x + y + z - 2 = 0,\; 2x + y - z - 3 = 0\) and \(3x + 2y + kz - 4 = 0\). For unique solution, \(\Delta \neq 0\). Find the value(s) of \(k\).

The value of θ lying between -π/4 & π/2 and 0 ≤ A ≤ π/2 and satisfying the equation 1+sin2 Acos2 A2sin 4θsin2 A1+cos2 A2sin 4θsin2 Acos2 A1+2sin 4θ = 0 are -

If \(f(x) = \begin{vmatrix} 1 & 2x & 3x^2 \\ x & x^2 & x^3 \\ 0 & 2 & 6x \end{vmatrix}\) then \(f'(1)\) = ______

Let $pq^3 + q q^3 + r z^2 + xz + t = \begin{vmatrix} x^2+3x-1 & x+3 \\ 2x & -2x & -4 \\ 3 & x & 3 \end{vmatrix}$ be an identity, where $p, q, r, s$ and $t$ are constants, then the value of $s$ is equal to

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

If x3x-yzx+z3y-w=3247, then

Let a, b, c, l, m, n ∈ R such that al + bm + cn = 0, bl + cm + an = 0, cl + am + bn = 0. If a, b & c are distinct & f(x) = ax^3 + bx^2 + cx + 5, then the value of f(1) is

Let $P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ and $X = \sum_{k=1}^6 P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?(A) $X - 30I$ is an invertible matrix(B) The sum of diagonal entries of $X$ is 18(C) If $X \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, then $\alpha = 30$(D) $X$ is a symmetric matrix

If AX = B where A is 3 x 3 and X and B are 3 x 1 matrices then which of the following is correct ?(A) If |A| = 0 then AX = B has infinite solutions(B) If AX = B has infinite solutions then |A| = 0(C) If (adj(A))B = 0 and |A| != 0 then AX = B has unique solution(D) If (adj(A))B != 0 & |A| = 0 then AX = B has no solution

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

The value of $\begin{vmatrix} 1 & 2 & 3 \\ -4 & 3 & 6 \\ 2 & -7 & 9 \end{vmatrix}$ is

The number of triplets (α, β, γ) satisfying the following constraints2α - β + 3γ = 4α + β - 3γ = -15α - β + 3γ = 7αβγ ≤ 0& α, β, γ ∈ I

Let 111123αβγ=t, where t is an even prime number & α, β, γ are the integral roots of the equation x3 - 14x2 + Px - 36 = 0On the basis of above information, answer the following :The value of P is -

Given the system of linear equations\(x - cy - cz = 0\)\(cx - y + cz = 0\)\(cx + cy - z = 0\)has a non-trivial solution, find the value of \(c\).

A $3 \times 3$ matrix has trace equal to 5 and is symmetric. How many such matrices are possible?

If $|A| = -15 + 14 = -1$, find $|A^{adj(A)} - |A^{20}|(A-3I)|$.

For the system $\begin{bmatrix} 1 & k & 3 \\ 2 & k & -3 \\ 3 & -4 & -2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ to have non-trivial solutions, find the value of $k$.

Let $A = \begin{bmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{bmatrix}$. If $A^2 = \begin{bmatrix} d_1^2 & 0 & 0 \\ 0 & d_2^2 & 0 \\ 0 & 0 & d_3^2 \end{bmatrix}$ and $kA = \begin{bmatrix} kd_1 & 0 & 0 \\ 0 & kd_2 & 0 \\ 0 & 0 & kd_3 \end{bmatrix}$, then find the total number of possible $3 \times 3$ matrices where each diagonal element is from $\{0, 1, -1\}$.

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 $P_1 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, P_3 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, P_4 = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}, P_5 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, P_6 = \begin{bmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{bmatrix}$ and $X = \sum_{k=1}^6 P_k \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1 \end{bmatrix} P_k^T$ where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?(A) $X - 30I$ is an invertible matrix(B) The sum of diagonal entries of $X$ is 18(C) If $X \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \alpha \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$, then $\alpha = 30$(D) $X$ is a symmetric matrix

Let S1 and S2 be respectively the sets of all a ∈ R - {0} for which the system of linear equations ax + 2ay - 3az = 1 (2a + 1)x + (2a + 3)y + (a + 1)z = 2 (3a + 5)x + (a + 5)y + (a + 2)z = 3 has unique solution and infinitely many solutions. Then

From the properties of invertible matrices, which of the following options is NOT true?

Let α and β be the distinct roots of the equation x2 + x - 1 = 0. Consider the set T = {1, α, β}. For a 3 × 3 matrix M = (aij)3×3, define Ri = ai1 + ai2 + ai3 and Cj = a1j + a2j + a3j for i = 1, 2, 3 and j = 1, 2, 3. Match each entry in List-I to the correct entry in List-II.List-I(P) The number of matrices M = (aij)3×3 with all entries in T such that Ri = Cj = 0 for all i, j, is(Q) The number of symmetric matrices M = (aij)3×3 with all entries in T such that Cj = 0 for all j, is(R) Let M = (aij)3×3 be a skew symmetric matrix such that aij ∈ T for i > j. Then the number of elements in the set { (x, y, z) ∈ R3 : M(x, y, z)T = 0 } is(S) Let M = (aij)3×3 be a matrix with all entries in T such that Ri = 0 for all i. Then the absolute value of the determinant of M isList-II(1) 1(2) 12(3) infinite(4) 6(5) 0

If a = α2 + β2 + γ2, b = αβ + βγ + γα, the value of \[\begin{vmatrix} b & a & b \\ b & a & b \\ b & b & a \end{vmatrix}\] is

Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:

The determinant xxz+xzyzzxx+y equals -

Let A, B and A + B are non-singular matrices of order 3 x 3 satisfying A^-1 + B^-1 = (A + B)^-1 and |AB^-1| is R then value of |A|/|B| 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

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

The value of \[\begin{vmatrix} \alpha & \beta & \gamma \\ \gamma & \alpha & \beta \\ \beta & \gamma & \alpha \end{vmatrix}\] where α, β, γ are the roots of x3 + 2x2 − x − 3 = 0 is equal to

For α, β ∈ R and a natural number n, let A_r = r1n22+α2r2n2-β3r-23n(3n-1)2. Then 2A_10 - A_8 is

Let M = \begin{pmatrix} \sin^2 \theta & -1-\sin^2 \theta \\ 1+\cos^2 \theta & \cos^2 \theta \end{pmatrix} = \alpha I + \beta M^{-1}, where \alpha = \alpha(\theta) and \beta = \beta(\theta) are real number, and I is the 2 \times 2 identity matrix. If \alpha^* is the minimum of the set \{\alpha(\theta): \theta \in [0, 2\pi]\} and \beta^* is the minimum of the set \{\beta(\theta): \theta \in [0, 2\pi]\}, then the value of \alpha^* + \beta^* is

For which of the following ordered pairs (μ, δ), the system of linear equations x + 2y + 3z = 1 3x + 4y + 5z = μ 4x + 4y + 4z = δ is inconsistent?

Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M-1 = adj(adj M), then which of the following statement is/are ALWAYS TRUE?