https://file.aiquickdraw.com/mathbot/file/1725886535367ykyo1u6z.jpg

The image contains two matrix-related exercises:

### Problem 1:
You're given the following matrices:

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

\[
D = \begin{bmatrix} 
1 & 3 \\ 
0 & 2 \\ 
-1 & 4 
\end{bmatrix}, \quad 
E = \begin{bmatrix} 
1 & 2 & 3 & -4 \\ 
2 & 0 & -2 & 1 
\end{bmatrix}, \quad 
F = \begin{bmatrix} 
1 & -2 & 3 \\ 
1 & 0 & -2 
\end{bmatrix}
\]

You are asked to compute the following:

1. \( A + B \)
2. \( B + (-1)C \)
3. \( (AD)E \)
4. \( A(DE) \)
5. \( D + F^T \)

### Problem 2:
The problem asks to show why, in general:

\[
(A \pm B)^2 \neq A^2 + 2AB + B^2 \quad \text{and} \quad (A + B)(A - B) \neq A^2 - B^2
\]

You're given matrices for \( A \) and \( B \):

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

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I can help with solving these matrix operations. Let's start with the first one: calculating \( A + B \). Should I proceed with that, or would you like to focus on a specific problem?

You're given the following matrices A, B, C, D, E, and F and asked to compute the sum, difference, and product of matrices, including transposes and other operations. The second part asks to show why certain matrix operations like (A ± B)^2 are not equal to A^2 + 2AB + B^2, given matrices A and B.

This problem explores key matrix operations including matrix addition, subtraction, multiplication, and transposition. Also, see why (A ± B)^2 is not equal to A^2 + 2AB + B^2 using example matrices A, B, C, D, E, and F. Ideal for undergraduate mathematics students.

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