https://file.aiquickdraw.com/mathbot/file/1727725772083pa08j5oy.jpg

The problem in the image asks to find a \( 2 \times 2 \) matrix \( C \) such that the composition of linear transformations, \( T_A \circ T_B \), equals \( T_C \). In terms of matrix multiplication, this means we need to find a matrix \( C \) such that:

\[
A \cdot B = C
\]

where:
\[
A = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}
\]

To solve for \( C \), we perform the matrix multiplication \( A \cdot B \).

Let's calculate it:

\[
C = A \cdot B = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix} \cdot \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}
\]

Now, I'll compute the product of these matrices.The matrix \( C \) that satisfies the equation \( T_A \circ T_B = T_C \) is:

\[
C = \begin{bmatrix} 3 & 4 \\ 10 & 12 \end{bmatrix}
\]

This matches the matrix \( C \) given in the problem, confirming that the answer is correct.

Would you like any further details or explanations?

Here are 5 related questions to expand on this topic:
1. How would the result change if matrix \( A \) was different?
2. What happens if we reverse the multiplication order, calculating \( B \cdot A \)?
3. How is matrix multiplication related to composition of linear transformations?
4. Can we find an inverse for matrix \( C \)? What would it represent geometrically?
5. What is the significance of the diagonal matrix \( A \) in this context?

**Tip:** In linear transformations, matrix multiplication represents the composition of two transformations, where the order of multiplication matters.

For any n × n matrix M, we define the linear transformation T_M(x) = Mx. Let n = 2 and matrices A and B be given. Find a 2 × 2 matrix C such that T_A ∘ T_B = T_C.

Solve a matrix multiplication problem to find a 2x2 matrix C such that T_A ∘ T_B equals T_C. This linear algebra problem involves linear transformations and matrix multiplication, suitable for undergraduate students.

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