Algebra of Matrices: JEE Questions & Solutions

For the matrices A = 3 -4 1 -1 and B = -29 49 -13 18, if (A¹⁵ + B) x y = 0 0, then among the following which one is true?

Let A = 0 2 -3 -2 0 1 3 -1 0 and B be a matrix such that B(I - A) = I + A. Then the sum of the diagonal elements of B^T B is equal to

The number of 3×2 matrices A, which can be formed using the elements of the set {-2,-1,0,1,2} such that the sum of all the diagonal…

Let A = [a_ij] be a 3 × 3 matrix such that A 0 1 0 = 0 0 1, A 4 1 3 = 0 1 0, A 2 1 2 = 1 0 0. Then a₂₃ equals:

Let A = [ 1/√2 -2 0 1 ] and P = [ cos θ -sin θ sin θ cos θ ], θ > 0. If B = P A P^T, C = P^T B¹⁰ P, and the sum of the diagonal elements of…

Let S = { m ∈ ℤ: A^m² + A^m = 3I - A⁻⁶ }, where A = 2 -1 1 0 Then n(S) is equal to:

Let A = [a_ij] be a matrix of order 3 × 3, with a_ij = (√2)^(i+j). If the sum of all the elements in the third row of A² is α + β√2, where…

Let A = α -1 6 β, α > 0, such that det(A) = 0 and α + β = 1. If I denotes the 2 × 2 identity matrix, then the matrix (I + A)⁸ is:

Let A be a 3 × 3 real matrix such that A²(A - 2I) - 4(A - I) = O, where I and O are the identity and null matrices, respectively. If A⁵ = α…

Let A = cos θ 0 -sin θ 0 1 0 sin θ 0 cos θ. If for some θ ∈ (0, π), A² = A^T, then the sum of the diagonal elements of the matrix ((A + I)³…

Let the matrix A = 1 0 0 1 0 1 0 1 0 satisfy Aⁿ = Aⁿ⁻² + A² - I for n ≥ 3. Then the sum of all the elements of A⁵⁰ is:

Given f(x)= cos x -sin x 0 sin x cos x 0 0 0 1 Evaluate: Statement I: f(-x) is the inverse of f(x). Statement II: f(x)· f(y)=f(x+y).

Let A be a 2 × 2 real matrix and I be the identity matrix of order 2. If the roots of the equation |A - xI| = 0 are -1 and 3, then the sum…

Let A be a square matrix such that AA^T = I. Then 1/2A[(A + A^T)² + (A - A^T)²] is equal to:

Let R = x 0 0 0 y 0 0 0 z be a non-zero 3 × 3 matrix, where xsinθ = ysin(θ + 2π/3) = zsin(θ + 4π/3) ≠ 0, θ ∈ (0, 2π). For a square matrix…

Let α and β be real numbers. Consider a 3 × 3 matrix A such that: A² = 3A + α I, A⁴ = 21A + β I. Then:

Let A be a symmetric matrix such that |A| = 2 and 2 1 3 3/2 A = 1 2 α β. If the sum of the diagonal elements of A is s, then (β s)/α² is…

Let A = 1 0 0 0 4 -1 0 12 -3 Then the sum of the diagonal elements of the matrix (A + I)¹¹ is equal to:

If A = 1/2 1 √3 -√3 1, then:

Let A = [a_ij]_(2× 2), where a_ij ≠ 0 for all i, j, and A² = I. Let a be the sum of all diagonal elements of A and b = |A|. Then 3a² + 4b²…

Let P be a square matrix such that P² = I - P. For α, β, γ, δ ∈ ℕ, if P^α + P^β = γ I - 29P and P^α - P^β = δ I - 13P then α + β + γ - δ is…

Let P = √3/2 1/2 -1/2 √3/2, A = 1 1 0 1, and Q = PAP^T. If P^TQ²⁰⁰⁷P = a b c d, then 2a + b - 3c - 4d equals:

Let A be a 2 × 2 matrix with real entries such that A^T = α A + I, where α ∈ ℝ \ {-1, 1}. If det(A² - A) = 4, then the sum of all possible…

Let A = 0 1 2 a 0 3 1 c 0, where a, c ∈ ℝ. If A³ = A and the positive value of a belongs to the interval (n-1, n], where n ∈ ℕ, then n is…

Let A = 1 1/51 0 1. If B = 1 2 -1 -1 A -1 -2 1 1, then the sum of all the elements of the matrix Σ_(n=1)⁵⁰ Bⁿ is equal to: