Matrix A Satisfies A²⁰+αA¹⁹+βA=Given Matrix — Find β−α
Given a triangular matrix A satisfying A²⁰+αA¹⁹+βA equals a specific matrix, find β−α using the eigenvalues on A’s diagonal.
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Sum of Prime Factors of |P⁻¹AP−2I| for Given 3×3 Matrices A, P
Given 3×3 matrices A and P, find the sum of the prime factors of |P⁻¹AP−2I| using the property that similar matrices share the same determinant after shifting by 2I.
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Aryan and Babban’s Monthly Incomes in Ratio 3:4, Solved Using Matrix Method
The monthly incomes of Aryan and Babban are in the ratio 3:4 and expenditures in the ratio 5:7. If each saves ₹15,000, find their incomes using the matrix method.
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Let A=[0 -2,2 0], M=ΣA²ᵏ, N=ΣA²ᵏ⁻¹ (k=1 to 10): Find MN²
Find MN² for a 2×2 matrix A where M and N are sums of its even and odd powers. Solved by spotting that A² is a scalar multiple of the identity.
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Let A, B=A−I, ω=(√3i−1)/2: Count n With Aⁿ+(ωB)ⁿ=A+B
Given A^n + (ωB)^n = A + B for ω a complex cube root of unity, how many n in 1..100 satisfy it? JEE Main 2022 matrix and complex numbers problem.