Adjoint of a Matrix

JEE Main Mathematics · Matrices and Determinants · 31 questions, latest first

• Let A= [α, 1, 2; 2, 3, 0; 0, 4, 5] and B= [1, 0, 0; 0, -5 α, 0; 0, 4 α, -2 α] +adj(A). If det(B)=66, then det(adj(A)) equals:20268 April, Shift 2 · Q4
• Let A= [-1, 1, -1; 1, 0, 1; 0, 0, 1] satisfy A^2+α(adj(adj(A)))+β(adj(A)(adj(adj(A))))= [2, -2, 2; -2, 0, -1; 0, 0, -1] for some…20266 April, Shift 1 · Q21
• Let A be a 3 × 3 matrix such that A^T [1; 0; 1] = [5; 2; 2], A^T [0; 0; 1] = [3; 1; 1], A [1; 0; 1] = [3; 4; 4] and A [0; 0; 1] =…20265 April, Shift 1 · Q3
• Let A= [1, 1, 2; -2, 0, 1; 1, 3, 5]. Then the sum of all elements of the matrix adj(adj(2(adj A)^-1)) is equal to:20264 April, Shift 1 · Q6
• 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)=2^10. Then the value of…202624 January, Shift 2 · Q4
• Let f(x)=∫ (7 x^10+9 x^8)/((1+x^2+2 x^9)^2) d x, x>0, lim_x → 0 f(x)=0 and f(1)=1/4. If A= [0, 0, 1; 1/4, f^′(1), 1; α^2, 4, 1]…202624 January, Shift 2 · Q18
• Let |A|=6, where A is a 3 × 3 matrix. If |adj(3 adj( A^2 · adj(2 A)))|=2^m · 3^n, m, n ∈ N, then m+n is equal to ____.202623 January, Shift 1 · Q21
• If X= [x; y; z] is a solution of the system of equations A X=B, where adj A= [4, 2, 2; -5, 0, 5; 1, -2, 3] and B= [4; 0; 2], then…202622 January, Shift 2 · Q6
• Let A= [2, 2+p, 2+p+q; 4, 6+2 p, 8+3 p+2 q; 6, 12+3 p, 20+6 p+3 q]. If det(adj(adj(3 A)))=2^m · 3^n, m, n ∈ N, then m+n is equal…20258 April, Shift 2 · Q4
• 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|^(3 n^2-5 n-4)}, then Σ_n ∈…20257 April, Shift 1 · Q5
• 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…20253 April, Shift 2 · Q22
• Let A be a matrix of order 3 × 3 and |A|=5. If |2 adj(3 A adj(2 A))|=2^α · 3^β · 5^γ, α, β, γ ∈ N, then α+β+γ is equal to20253 April, Shift 1 · Q5
• 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…20252 April, Shift 1 · Q15
• If A, B, and (adj(A^-1)+adj(B^-1)) are non-singular matrices of same order, then the inverse of A(adj(A^-1)+adj(B^-1))^-1 B, is…202523 January, Shift 1 · Q5
• 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…202522 January, Shift 2 · Q5
• Let A be a square matrix of order 3 such that det(A)=-2 and det(3 adj(-6 adj(3 A)))=2^m+n · 3^mn, m>n. Then 4 m+2 n is equal to…202522 January, Shift 1 · Q21
• Let A be a non-singular matrix of order 3. If det(3 adj(2 adj((det A) A)))=3^-13 · 2^-10 and det(3 adj(2 A))=2^m · 3^n, then |3…20249 April, Shift 1 · Q23
• If A is a square matrix of order 3 such that det(A)=3 and det(adj(-4 adj(-3 adj(3 adj((2 A)^-1)))))=2^m 3^n, then m+2 n is equal…20246 April, Shift 2 · Q4
• Let A and B be two square matrices of order 3 such that |A|=3 and |B|=2. Then |A^T A(adj(2 A))^-1(adj(4 B))(adj(AB))^-1 AA^T| is…20245 April, Shift 1 · Q3
• Let α ∈(0, ∞) and A= [1, 2, α; 1, 0, 1; 0, 1, 2]. If det(adj(2 A-A^T) · adj(A-2 A^T))=2^8, then (det(A))^2 is equal to:20244 April, Shift 1 · Q5
• Let A be a 3 × 3 matrix and det(A)=2. If n=det( adj(adj(… ….(adj A))))_2024- times, then the remainder when n is divided by 9 is…202431 January, Shift 2 · Q22
• Let the determinant of a square matrix A of order m be m-n, where m and n satisfy 4 m+n=22 and 17 m+4 n=93. If det(n adj(adj(m…202315 April, Shift 1 · Q4
• Let for A= [1, 2, 3; α, 3, 1; 1, 1, 2],|A|=2. If |2 adj(2 adj(2 A))|=32^n, then 3 n+α is equal to202313 April, Shift 2 · Q5
• Let B= [1, 3, α; 1, 2, 3; α, α, 4], α>2 be the adjoint of a matrix A and |A|=2. Then [α, -2 α, α] B [α; -2 α; α] is equal to202313 April, Shift 1 · Q3
• If A=1/(5!6!7!) [5!, 6!, 7!; 6!, 7!, 8!; 7!, 8!, 9!], then |adj(adj(2 A))| is equal to202310 April, Shift 2 · Q4
• If A is a 3 × 3 matrix and |A|=2, then |3 adj(|3 A| A^2)| is equal to202310 April, Shift 1 · Q4
• Let A= [2, 1, 0; 1, 2, -1; 0, -1, 2]. If |adj(adj(adj 2 A))|=(16)^n, then n is equal to20238 April, Shift 1 · Q4
• Let A be a n × n matrix such that |A|=2. If the determinant of the matrix Adj(2 · Adj(2 A^-1)) · is 2^84, then n is equal to ____.202331 January, Shift 2 · Q81
• Let A= (m, n; p, q), d=|A| ≠ 0 and |A-d(Adj A)|=0. Then202330 January, Shift 1 · Q63
• Let x, y, z>1 and A= [1, log_x y, log_x z; log_y x, 2, log_y z; log_z x, log_z y, 3]. Then |adj(adj A^2)| is equal to202325 January, Shift 1 · Q63
• Let A be a 3 × 3 matrix such that |adj(adj(adj A))|=12^4 Then |A^-1 adj A| is equal to202324 January, Shift 2 · Q65