Matrices and Determinants

13 solved problems from JEE previous year papers.

Determinants

If the Determinant |x−2, 2x−3, 3x−4; 2x−3, 3x−4, 4x−5; 3x−5, 5x−8, 10x−17| = Ax³ + Bx² + Cx + D, Then B + C Is Equal To

JEE Main 2018 PYQ: expand a 3×3 determinant in x as Ax³+Bx²+Cx+D using column operations to create zeros, then find B+C.
Matrices

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))⁻¹B Is Equal To

Given A, B, and adj(A⁻¹)+adj(B⁻¹) are non-singular matrices of the same order, find the inverse of A[adj(A⁻¹)+adj(B⁻¹)]⁻¹B using…
Matrices

If A = [cos θ, i sin θ; i sin θ, cos θ], θ = π/24 and A⁵ = [a b; c d], Where i = √−1, Then Which One of the Following Is Not True

For A=[cosθ isinθ;isinθ cosθ] with θ=π/24 and A⁵=[a b;c d], determine which of four given statements about a,b,c,d is not true,…
CBSE All India 2022

If x = −4 Is a Root of |x 2 3; 1 x 1; 3 2 x| = 0, Then Find the Sum of the Other Two Roots

Given x=−4 satisfies the determinant equation |x 2 3; 1 x 1; 3 2 x|=0, expand it into a cubic and find the sum of the remaining…
Matrices and Determinants

Let A = [[1,−2,α],[α,2,−1]] and B = [[2,α],[−1,2],[4,−5]], α ∈ C. Then the Absolute Value of the Sum of All Values of α for Which det(AB) = 0 Is

For A=[1,−2,α;α,2,−1] and B=[2,α;−1,2;4,−5], find the absolute value of the sum of all complex α for which det(AB)=0, by…
Determinants

Determinant Dk Sum From k=1 to n Equals 96, Find n

A JEE Main 2023 problem: sum the determinant D_k of a given 3×3 matrix from k=1 to n, set the total to 96, and solve for n.
Matrices

2×2 Matrix A With A’=αA+I and det(A²−A)=4, Sum of All Possible α

Let A be a 2×2 matrix with real entries such that A’=αA+I, α∈R−{−1,1}. If det(A²−A)=4, find the sum of all possible values…
Matrices and Determinants

If A and B Are Symmetric Matrices of the Same Order, Show That AB Is Symmetric if and Only if AB = BA

Prove that for symmetric matrices A and B of the same order, the product AB is symmetric if and only if AB=BA, using the…
Matrices and Determinants

If the Matrix A = [[1,0,0],[0,2,0],[3,0,−1]] Satisfies A²⁰ + αA¹⁹ + βA = [[1,0,0],[0,4,0],[0,0,1]] for Real Numbers α and β, Then β − α Is Equal To

Given a triangular matrix A satisfying A²⁰+αA¹⁹+βA equals a specific matrix, find β−α using the eigenvalues on A’s diagonal.
Matrices and Determinants

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

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…
CBSE Class 12

The Monthly Incomes of Aryan and Babban Are in the Ratio 3:4 and Their Monthly Expenditures Are in the Ratio 5:7. If Each Saves ₹15,000 per Month, Find Their Monthly Incomes Using the 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…
Matrices

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…
Matrices

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…