If `2A+B=[(0,0,3),(10,1,9),(-1,4,0)] and A-2B=[(0,-5,9),(-5,3,2),(-3,2,0)]` Find the value of `(t_r(A)-t_r(B))`

Options

A. 1
B. -2
C. -1
D. 2

Answer

To solve the problem, we need to find the value of \( t_r(A) - t_r(B) \) given the equations involving matrices \( A \) and \( B \).

### Step-by-step Solution:

1. **Write the Given Equations:**
We have two equations:
\[ 2A + B = \begin{pmatrix} 0 & 0 & 3 \\ 10 & 1 & 9 \\ -1 & 4 & 0 \end{pmatrix} \]
\[ A - 2B = \begin{pmatrix} 0 & -5 & 9 \\ -5 & 3 & 2 \\ -3 & 2 & 0 \end{pmatrix} \]

2. **Take the Trace of Both Sides:**
The trace of a matrix is the sum of its diagonal elements. We will apply the trace operation to both equations.
- For the first equation:
\[ t_r(2A + B) = t_r\left(\begin{pmatrix} 0 & 0 & 3 \\ 10 & 1 & 9 \\ -1 & 4 & 0 \end{pmatrix}\right) \]
The trace of the right-hand side is:
\[ 0 + 1 + 0 = 1 \]
Using the property of trace, we have:
\[ t_r(2A) + t_r(B) = 1 \]
Since \( t_r(2A) = 2 \cdot t_r(A) \), we can rewrite this as:
\[ 2t_r(A) + t_r(B) = 1 \quad \text{(Equation 1)} \]

- For the second equation:
\[ t_r(A - 2B) = t_r\left(\begin{pmatrix} 0 & -5 & 9 \\ -5 & 3 & 2 \\ -3 & 2 & 0 \end{pmatrix}\right) \]
The trace of the right-hand side is:
\[ 0 + 3 + 0 = 3 \]
Using the property of trace, we have:
\[ t_r(A) - 2t_r(B) = 3 \quad \text{(Equation 2)} \]

3. **Solve the System of Equations:**
Now we have a system of two equations:
- \( 2t_r(A) + t_r(B) = 1 \) (Equation 1)
- \( t_r(A) - 2t_r(B) = 3 \) (Equation 2)

We can solve this system. From Equation 1, we can express \( t_r(B) \):
\[ t_r(B) = 1 - 2t_r(A) \]
Substitute \( t_r(B) \) into Equation 2:
\[ t_r(A) - 2(1 - 2t_r(A)) = 3 \]
Simplifying this gives:
\[ t_r(A) - 2 + 4t_r(A) = 3 \]
\[ 5t_r(A) - 2 = 3 \]
\[ 5t_r(A) = 5 \]
\[ t_r(A) = 1 \]

4. **Find \( t_r(B) \):**
Substitute \( t_r(A) \) back into the equation for \( t_r(B) \):
\[ t_r(B) = 1 - 2(1) = 1 - 2 = -1 \]

5. **Calculate \( t_r(A) - t_r(B) \):**
Now we can find the value of \( t_r(A) - t_r(B) \):
\[ t_r(A) - t_r(B) = 1 - (-1) = 1 + 1 = 2 \]

### Final Answer:
\[ t_r(A) - t_r(B) = 2 \]