Properties of Determinants: JEE Questions & Solutions

If A = 2 3 3 5, then the determinant of the matrix A²⁰²⁵ - 3A²⁰²⁴ + A²⁰²³ is

Let |A| = 6, where A is a 3 × 3 matrix. If |adj(3adj(A² · adj(2A)))| = 2^m · 3ⁿ, then m + n is equal to:

Let f(x)=∫ (7x¹⁰+9x⁸)/(1+x²+2x⁹)² dx, x>0, lim_(x→ 0)f(x)=0, f(1)=1/4. If A= 0 0 1 1/4 f'(1) 1 α² 4 1 and B=adj(adj A) is such that |B|=81,…

Let A be a square matrix of order 3 such that det(A) = -2 and det(3 · adj(-6 · adj(3A))) = 2^(m+n) · 3^mn, where m > n. Then 4m + 2n is…

Let A = [ a_ij ] = [ log₅ 128 log₄ 5 log₅ 8 log₄ 25 ]. If A_ij is the cofactor of a_ij, C_ij = Σ_(k=1)² a_ik A_jk, and C = [C_ij], then…

Let a ∈ ℝ and A be a matrix of order 3 × 3 such that det(A) = -4 and A + I = [ 1 a 1 2 1 0 a 1 2 ] where I is the identity matrix of order…

Let A = 2 1 2 6 2 11 3 3 2 and P = 1 2 0 5 0 2 7 1 5. The sum of the prime factors of |P⁻¹AP - 2I| is equal to:

If A = √2 1 -1 √2, B = 1 0 1 1, C = ABA^T, and X = A^T C² A, then det(X) is equal to:

Let A be a 3 × 3 matrix such that |adj(adj(adj A))| = 12⁴. Then |A⁻¹ adj A| is equal to:

Let f(x) = 1 + sin² x cos² x sin 2x sin² x 1 + cos² x sin 2x sin² x cos² x 1 + sin 2x where x ∈ [ π/6, π/3 ]. If α, β are respectively the…

If A is a 3×3 matrix and |A| = 2, then |3adj(|3A|A²)| is equal to: