Question

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 :

• A. 1
• B. 2
• C. 3
• D. 0

Hint:

Trace of a matrix is the sum of its eigenvalues and also the sum of its main diagonal elements.

Answer 1

The Correct Option is B

Step 1: Use the property that $Tr$ is a linear operator: $Tr(A+2B) = Tr(A) + 2Tr(B)$ and $Tr(2A-B) = 2Tr(A) - Tr(B)$.
Step 2: $Tr(A+2B) = 1 + (-3) + 1 = -1$. So, $Tr(A) + 2Tr(B) = -1$.
Step 3: $Tr(2A-B) = 2 + (-1) + 2 = 3$. So, $2Tr(A) - Tr(B) = 3$.
Step 4: Solve the system: Multiply first eq by 2: $2Tr(A) + 4Tr(B) = -2$. Subtract the second eq from this: $5Tr(B) = -5 \implies Tr(B) = -1$.
Step 5: Substitute back: $Tr(A) + 2(-1) = -1 \implies Tr(A) = 1$.
Step 6: $Tr(A) - Tr(B) = 1 - (-1) = 2$.