Inverse & Adjoint of a Matrix: JEE Questions & Solutions

For some α, β ∈ R, let A = α 2 1 2 and B = 1 1 1 β be such that A²-4A+2I = B²-3B+I = O. Then (det(adj(A³-B³)))² is equal to:

Let A be a 3 × 3 matrix such that A + A^T = O. If A 1 -1 0 = 3 3 2, A² 1 -1 0 = -3 19 -24 and det (adj(2 adj(A+I))) = 2^α 3^β 11^γ where α,…

If X= x y z is a solution of the system of equations AX=B, where adj A= 4 2 2 -5 0 5 1 -2 3 and B= 4 0 2, then |x+y+z| is equal to

Let P=[p_ij] and Q=[q_ij] be two square matrices of order 3 such that q_ij=2^(i+j-1)p_ij and det(Q)=2¹⁰. Then the value of det(adj(adj P))…

Let A, B, C be three 2×2 matrices with real entries such that B=(I+A)⁻¹ and A+C=I. If BC= 1 -5 -1 2 and CB x₁ x₂ = 12 -6, then x₁+x₂ is:

For a 3 × 3 matrix M, let trace(M) denote the sum of all the diagonal elements of M. Let A be a 3 × 3 matrix such that |A| = 1/2 and…

If A, B, and ( adj(A⁻¹) + adj(B⁻¹)) are non-singular matrices of the same order, then the inverse of A ( adj(A⁻¹) + adj(B⁻¹))⁻¹ B is equal…

Let A be a 3 × 3 matrix such that X^T A X = O for all nonzero 3 × 1 matrices X = x y z. If A 1 1 1 = 1 4 -5, A 1 2 1 = 0 4 -8 and…

Let A be a matrix of order 3 × 3 and |A| = 5. If |2 adj(3A adj(2A))| = 2^α · 3^β · 5^γ, α, β, γ ∈ ℕ then α + β + γ is equal to

Let I be the identity matrix of order 3 × 3 and for the matrix A = λ 2 3 4 5 6 7 -1 2, |A| = -1. Let B be the inverse of the matrix adj(A ·…

Let A be a 3 × 3 matrix such that | adj(adj(adj A)) | = 81. If S = { n ∈ ℤ: | adj(adj A) |^((n - 1)²/2) = |A|^(3n² - 5n - 4) }, then the…

Let A = 2 2 + p 2 + p + q 4 6 + 2p 8 + 3p + 2q 6 12 + 3p 20 + 6p + 3q If det(adj(adj(3A))) = 2^m · 3ⁿ, m, n ∈ ℕ, then m + n is equal to:

Let A = 1 2 0 1 and B = I + adj(A) + (adj(A))² +... + (adj(A))¹⁰. The sum of all elements in B is:

Let B = 1 3 1 5 and A be a 2 × 2 matrix such that AB⁻¹ = A⁻¹. If BCB⁻¹ = A and C⁴ + α C² + β I = O, then 2β - α is equal to:

Let x, y, z > 1 and A = 1 logₓ y logₓ z log_y x 2 log_y z log_z x log_z y 3. Then adj(adj A²) is equal to:

Let, then the inverse of the matrix A M²⁰²³ A^T is:

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

Let A = 2 1 0 1 2 -1 0 -1 2. If | adj(adj(adj(2A))) | = (16)ⁿ, then n is equal to:

Let S = { M = [a_ij], a_ij ∈ {0, 1, 2}, 1 ≤ i, j ≤ 2 } be a sample space and A = { M ∈ S: M is invertible } be an event. Then P(A) is equal…

Let B= 1 3 α 1 2 3 α α 4, α>2 be the adjoint of a matrix A with |A|=2. Then α -2α α B α -2α α is equal to:

Let for A = 1 2 3 α 3 1 1 1 2, |A| = 2. If |2 adj(2 adj(2A))| = 32ⁿ, then 3n + α is equal to:

Let the determinant of a square matrix A of order m be m-n, where m and n satisfy 4m+n=22 and 17m+4n=93. If det (n adj(adj(mA)))=3^a5^b6^c,…