JEE Main 2021 · 18 March · morning · Single correct · +4 / −1

Let A + 2B = [ 1, 2, 0; 6, - 3, 3; - 5, 3, 1 ] and 2A - B = [ 2, - 1, 5; 2, - 1, 6; 0, 1, 2 ]. If Tr(A) denotes the sum of all diagonal elements of the matrix A, then Tr(A) - Tr(B) has value equal to

  1. (a) 1
  2. (b) 2
  3. (c) 0
  4. (d) 3
Answer and solution

Answer (b) 2

Solution

Maths sir · 4 steps

Do not find A and B at all. Take the trace of both equations and solve for two numbers.

  1. Trace is linear. For any matrices P, Q and scalar k, Tr(P + kQ) = Tr(P) + k Tr(Q). So write a = Tr(A) and b = Tr(B), and you only need two numbers, not two whole matrices.
  2. Read the diagonals. The diagonal of A + 2B is 1, -3, 1, so a + 2b = -1. The diagonal of 2A - B is 2, -1, 2, so 2a - b = 3.

    a + 2b = -1 ; 2a - b = 3

  3. Solve the pair. From the second equation, b = 2a - 3. Substituting gives a + 4a - 6 = -1, so 5a = 5, a = 1 and b = -1.
  4. So. Tr(A) - Tr(B) = 1 - (-1) = 2, which is option (b).

The idea to keep. When a question asks only for traces, take the trace of every given equation. Trace is linear, so the matrices shrink to ordinary numbers and a simple linear system does the rest.

Answer from the NTA final answer key. Solution by PrepWiser.

Study Matrices

Asked about 110 times in JEE Main since 2020. This question is from 2021.