Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
99 Percentile Qs Bank for JEE MainMathematicsMatrices

The number of matrices X with entries 0 , 2,3 for which the sum of all the principal diagonal elements of X · X T is 28 (where X T is the transpose matrix of X ), is

Options

  1. A12
  2. B18
  3. C36
  4. D44

Correct answer

C. 36

Step-by-step solution

Let X = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 X X T = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 a 1 b 1 c 1 a 2 b 2 c 2 a 1 b 2 c 3 = a 1 2 + a 2 2 + a 3 2 a 1 b 1 + a 2 b 2 + a 3 b 3 a 1 c 1 + a 2 c 2 + a 3 c 3 a 1 b 1 + a 2 b 2 + a 3 b 3 b 1 2 + b 2 2 + b 3 2 b 1 c 1 + b 2 c 2 + b 3 c 3 a 1 c 1 + a 2 c 2 + a 3 c 3 b 1 c 1 + b 2 c 2 + b 3 c 3 c 1 2 + c 2 2 + c 3 2 T r X X T = a 1 2 + a 2 2 + a 3 2 + b 1 2 + b 2 2 + b 3 2 + c 1 2 + c 2 2 + c 3 2 = 28 There is only one possibility in which 7 entries are 2 and 2 entries are

Practice Matrices on Quantrex Academy →

More from Matrices

Which one of the following matrices can be obtained by performing elementary row transformations on the 3 3 identity matrix? 2026Consider the matrix M = bmatrix 2 & -1 1 & 0 bmatrix . Let p, q, r, s, a, b, c and d be integers such that M²⁶ = bmatrix p & q r & s bmatrix and _ k=1 ²⁶ M^k = bmatrix a & b c & d 2026For real numbers , , , and , consider the matrix M = bmatrix & 1 2 & - 1 2 1 3 & & 1 3 & & bmatrix . Suppose that MM^T = I , where M^T is the transpose of the matrix M , and I is t 2026Let R denote the set of all real numbers and let i = -1 . Consider the matrices S = bmatrix 0 & -1 1 & 0 bmatrix and T = bmatrix 1 & 1 0 & 1 bmatrix . Let a, b, c, d be real number 2026Let A = bmatrix & 1 & 2 2 & 3 & 0 0 & 4 & 5 bmatrix and B = bmatrix 1 & 0 & 0 0 & -5 & 0 0 & 4 & -2 bmatrix + adj (A) . If (B)=66 , then ( adj (A)) equals: 2026Let A = bmatrix 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 bmatrix and B = [b_ ij ] , 1 i, j 3 . If B = A⁹⁹ - I , then the value of b₃₁ - b₂₁ b₃₂ is : 2026Let A = bmatrix -1 & 1 & -1 1 & 0 & 1 0 & 0 & 1 bmatrix satisfy A^2 + (adj(adj(A))) + (adj(A)(adj(adj(A)))) = bmatrix 2 & -2 & 2 -2 & 0 & -1 0 & 0 & -1 bmatrix for some , R . Then 2026Let M be a 3 3 matrix such that M pmatrix 1 0 0 pmatrix = pmatrix 1 2 3 pmatrix , M pmatrix 0 1 0 pmatrix = pmatrix 0 1 2 pmatrix and M pmatrix 0 0 1 pmatrix = pmatrix -1 1 1 pmatr 2026 Full Matrices list All 99 Percentile Qs Bank for JEE Main PYQs