Matrices & Determinants Questions (2045)

Let a determinant is given by A = abcpqrxyz and suppose A = 6. If B = p+xq+yr+za+xb+yc+za+pb+qc+r then

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices : Tp = {A = abca : a, b, c ∈ {0, 1, 2, ..., p-1}}. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is -

Let A = 2-111. If the sum of the diagonal elements of A13 is 3n, then n is equal to ____.

If px^4 + qx^3 + rx^2 + sx + t = \[ \begin{vmatrix} \(x^2+3x\) & \(x-1\) & \(x+3\) \\ \(x+1\) & \(2-x\) & \(x-3\) \\ \(x-3\) & \(x+4\) & 3x \end{vmatrix} \], then t is equal to:

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

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 ____.

Let A be a 2 x 2 matrix with non-zero entries and let A2 = I, where I is 2 x 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement-1 : Tr(A) = 0.Statement-2 : |A| = 1.(A) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.(B) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for statement-1.(C) Statement-1 is true, Statement-2 is false.(D) Statement-1 is false, Statement-2 is true.

If A and B are symmetric matrices and AB = BA, then A-1B is a -

Consider the system of equations : x + ay = 0, y + az = 0 and z + ax = 0. Then the set of all real values of 'a' for which the system has a unique solution is :

The system of linear equations x + y + z = 6, x + 2y + 3z = 14 and 2x + 5y + pz = q have -

If A is a symmetric and B skew symmetric matrix and A + B is non singular and C = (A + B)-1(A - B) then CT AC =

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

The system of linear equations x + λy - z = 0, λx - y - z = 0, x + y - λz = 0 has a non-trivial solution for :

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

If A = 1000110-24, I = 100010001 and A-1 = 16(A2 + cA + dI), then the value of c + d is:

If px4 + qx3 + rx2 + sx + t = x2+3xx-1x+3x+12-xx-3x-3x+43x then t is equal to -

The total number of distinct x ∈ R for which <mfenced open="|

The determinant (x2+1)2(xy+1)2(xz+1)2(xy+1)2(y2+1)2(yz+1)2(xz+1)2(yz+1)2(z2+1)2 = k(x-y)^2(y-z)^2(z-x)^2, then k =

Let f(x) = <mfenced open="|

If S_r = 2rxn(n+1)6r2-1yn2(2n+3)4r3-2nrzn3(n+1), then ∑r=1nSr does not depend on -

Which of following statement is/are false -(A) t is divisible by (α - β)(B) t is divisible by (β - γ)(C) t is divisible by (γ - α)(D) (γ - α) is divisible by t

The values of α, for which 13/2α+3/211/3α+1/32α+33α+10 = 0, lie in the interval [JEE (Main) 2024]

Let P = 3-1-220α3-50, where α ∈ R. Suppose Q = [qij] is a matrix such that PQ = kI, where k ∈ R, k ≠ 0 and I is the identity matrix of order 3. If q23 = -k/8 and det(Q) = k2/2, then(A) α = 0, k = 8(B) 4α - k + 8 = 0(C) det(P adj(Q)) = 29(D) det(Q adj(P)) = 213

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 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-

The number of 3 x 3 matrices A whose entries are either 0 or 1 and for which the system A[x y z]^T = [1 0 0]^T has exactly two distinct solutions is:

Sum of elements of (adjA) B is -

Let A = 2-111. If the sum of the diagonal elements of A13 is 3n, then n is equal to ____.

Number of real values of λ for which the matrix A = $\begin{bmatrix} \lambda-1 & \lambda & \lambda+1 \\ 2 & -1 & 3 \\ \lambda+3 & \lambda-2 & \lambda+7 \end{bmatrix}$ has no inverse

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2x2 matrix such that the trace of A is 3 and the trace of A^3 is -18, then the value of the determinant of A 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

For the system of linear equation2x - y + 3z = 53x + 2y - z = 74x + 5y + αz = βWhich of the following is(are) CORRECT?(A) The system has infinitely many solutions for α = -5 and β = 9(B) The system has a unique solution for α ≠ -5 and β = 8(C) The system has infinitely many solutions for α = -6 and β = 9(D) The system is inconsistent for α = -5 and β = 8

Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is

If determinant \(\Delta\) is such that one row is all ones and others arbitrary, then \(\Delta\) simplifies by:

Let d ∈ R, and A = -24+dsinθ-21sinθ+2d52sinθ-d-sinθ+2+2d, θ ∈ [0, 2π]. If the minimum value of det(A) is 8, then a value of d is :

Consider the following statementsStatement-1 : Given $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 4 & 1 \\ 2 & 3 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 3 \\ 3 & 4 \end{bmatrix}$. If $BPA = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then $\lambda$ denotes sum of elements of $P$.Statement-2 : Let $\mu$ denote the sum of elements of the matrix $A$ satisfying the matrix equation, $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} 3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 3 & -1 \end{bmatrix}$Statement-3 : Given that $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 2 & 1 \\ 1 & -1 & 3 \end{bmatrix}$, $C = \begin{bmatrix} 2 & 1 & 1 \\ 2 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 10 \\ 13 \\ 9 \end{bmatrix}$ and that $Cb = D$. If $AX = b$, then $v$ denotes the sum of elements of $X$.Then, which of the following options is/are correct?(A) $2\lambda - 19\mu = 3$(B) $38\mu + 15v = 3$(C) $10v + 8\mu = 2$(D) $\lambda + 19\mu + 38\mu = 0$

If a, b, c > 0 and x, y, z ∈ R, then the determinant <mfenced close="|

Given: \(x + ky + 3z = 0,\; 3x + ky - 2z = 0,\; 2x + 4y - 3z = 0\)For non-zero solutions, find the value of \(k\). Hence find \(\dfrac{xz}{y^2}\).

If P is a 3 x 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 x 3 identity matrix, then there exists a column matrix X = xyz ≠ 000 such that

Let p, q, r be nonzero real numbers that are, respectively, the 10th, 100th and 1000th terms of a harmonic progression. Consider the system of linear equationsx + y + z = 110x + 100y + 1000z = 0qrx + pry + pqz = 0.

If A = 1234, then A2 - 5A - 2I is equal to

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:

The determinant , then k =

Let \(A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\). Then \(A^n\) equals

Let a determinant is given by A = abcpqrxyz and suppose A = 6. If B = p+xq+yr+za+pb+qc+rabc then

If the matrices A = 1121341-13, B = adj A and C = 3A, then |adj B|/|C| is equal to :

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

5. $\begin{vmatrix} a_1+b_1x & a_1x+b_1 & c_1 \\ a_2+b_2x & a_2x+b_2 & c_2 \\ a_3+b_3x & a_3x+b_3 & c_3 \end{vmatrix} = 0$, then possible conditions is/are -

Match the following: A -> S; B -> R; C -> P; D -> Q

Let A be a 3x3 matrix such that A^2 = I. If A is not equal to I and A is not equal to -I, then which of the following is true?