The given problem asks us to find the trace of a matrix that results from multiplying two matrices:

$A = \begin{bmatrix} 1 & 2 \\ -2 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$

First, let me multiply these two matrices $A$ and $B$:

Step 1: Matrix Multiplication $A \times B$

The product of matrices $A$ and $B$, denoted as $C = A \times B$, is calculated as follows:

$C = \begin{bmatrix} 1 & 2 \\ -2 & 2 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$

To multiply these matrices, we calculate each element $c_{ij}$ of the resulting matrix $C$:

• $c_{11} = (1 \cdot 2) + (2 \cdot 0) = 2$
• $c_{12} = (1 \cdot 1) + (2 \cdot 3) = 1 + 6 = 7$
• $c_{21} = (-2 \cdot 2) + (2 \cdot 0) = -4$
• $c_{22} = (-2 \cdot 1) + (2 \cdot 3) = -2 + 6 = 4$

Thus, the product matrix $C$ is:

$C = \begin{bmatrix} 2 & 7 \\ -4 & 4 \end{bmatrix}$

Step 2: Calculate the Trace

The trace of a matrix is the sum of the elements on the main diagonal. For matrix $C$:

$\text{Trace}(C) = c_{11} + c_{22} = 2 + 4 = 6$

Final Answer

The trace of the resulting matrix is:

$\boxed{6}$

Do you want any additional details or have further questions?

Here are some related questions that may expand your understanding:

1. What are the properties of the trace of a matrix?
2. How do you find the determinant of a product of two matrices?
3. Can you explain the significance of the trace in linear algebra?
4. What happens to the trace if we transpose the resulting matrix?
5. How does matrix multiplication affect eigenvalues and their sum?

Tip: The trace of a matrix is invariant under transposition, meaning $\text{Tr}(A) = \text{Tr}(A^T)$. This can be useful in simplifying computations involving traces.