Let A + 2B = $\begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and 2A - B = $\begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$, then Tr(A) - Tr(B) has the value equal to
Step-by-Step Solution
Key Concept: Use the properties of the trace operator: Tr(A + 2B) = Tr(A) + 2Tr(B) and Tr(2A - B) = 2Tr(A) - Tr(B). Calculate the trace of the given matrices and solve the system of linear equations for Tr(A) and Tr(B).
Let $T_1 = \text{Tr}(A+2B) = 1 - 3 + 1 = -1$. Thus, $\text{Tr}(A) + 2\text{Tr}(B) = -1$. Let $T_2 = \text{Tr}(2A-B) = 2 - 1 + 2 = 3$. Thus, $2\text{Tr}(A) - \text{Tr}(B) = 3$. Multiplying the second equation by 2 gives $4\text{Tr}(A) - 2\text{Tr}(B) = 6$. Adding this to the first equation gives $5\text{Tr}(A) = 5$, so $\text{Tr}(A) = 1$. Substituting back, $2(1) - \text{Tr}(B) = 3$, so $\text{Tr}(B) = -1$. Therefore, $\text{Tr}(A) - \text{Tr}(B) = 1 - (-1) = 2$.
Correct Answer: 3