Powers of a Matrix and Matrix Polynomials

JEE Main Mathematics · Matrices and Determinants · 16 questions, latest first

• Let A= [1, 0, 0; 3, 1, 0; 9, 3, 1] and B=[b_i j], 1 ≤ i, j ≤ 3. If B=A^99-I, then the value of (b_31-b_21)/(b_32) is:20266 April, Shift 2 · Q5
• Let A= [3, -4; 1, -1] and B be two matrices such that A^100=100 B+I. Then the sum of all the elements of B^100 is ____202628 January, Shift 2 · Q21
• For the matrices A= [3, -4; 1, -1] and B= [-29, 49; -13, 18], if (A^15+B) [x; y] = [0; 0], then among the following which one is…202621 January, Shift 2 · Q5
• Let the matrix A= [1, 0, 0; 1, 0, 1; 0, 1, 0] satisfy A^n=A^n-2+A^2-I for n ≥ 3. Then the sum of all the elements of A^50 is:20254 April, Shift 2 · Q4
• Let A= [cos θ, 0, - sin θ; 0, 1, 0; sin θ, 0, cos θ]. If for some θ ∈(0, π), A^2=A^T, then the sum of the diagonal elements of…20254 April, Shift 1 · Q22
• Let A be a 3 × 3 real matrix such that A^2(A-2 I)-4(A-I)=O, where I and O are the identity and null matrices, respectively. If…20252 April, Shift 2 · Q3
• Let A= [α, -1; 6, β], α>0, such that det(A)=0 and α+β=1. If I denotes 2 × 2 identity matrix, then the matrix (I+A)^8 is:20252 April, Shift 1 · Q18
• Let S={m ∈ Z: A^m^2+A^m=3 I-A^-6}, where A= [2, -1; 1, 0]. Then n(S) is equal to ____.202529 January, Shift 1 · Q21
• Let A= [1/(√2), -2; 0, 1] and P= [cos θ, - sin θ; sin θ, cos θ], θ>0. If B=PAP^T, C=P^T B^10 P and the sum of the diagonal…202528 January, Shift 2 · Q5
• Let A= [2, -1; 1, 1]. If the sum of the diagonal elements of A^13 is 3^n, then n is equal to ____.20248 April, Shift 1 · Q22
• Let A= [1, 2; 0, 1] and B=I+adj(A)+(adj A)^2+…+(adj A)^10. Then, the sum of all the elements of the matrix B is:20244 April, Shift 2 · Q3
• 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 B^n is…202312 April, Shift 1 · Q3
• Let A= [0, 1, 2; a, 0, 3; 1, c, 0], where a, c ∈ R. If A^3=A and the positive value of a belongs to the interval ( n-1, n ],…202311 April, Shift 1 · Q29
• Let P be a square matrix such that P^2=I-P. For α, β, γ, δ ∈ N, if P^α+P^β=γ I-29 P and P^α-P^β=δ I-13 P, then α+β+γ-δ is equal to20236 April, Shift 2 · Q4
• 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:202331 January, Shift 1 · Q64
• Let α and β be real numbers. Consider a 3 × 3 matrix A such that A^2=3 A+α I. If A^4=21 A+β I, then202329 January, Shift 1 · Q66