https://file.aiquickdraw.com/mathbot/file/1726215257611lwgoxi7t.jpg

Let's solve the problems given in the image step by step. We are provided with three matrices \( A \), \( B \), and \( C \):

\[
A = \begin{pmatrix} 2 & 1 & 0 & 3 \\ -1 & 0 & 2 & 4 \\ 4 & -2 & 7 & 0 \end{pmatrix}
\]

\[
B = \begin{pmatrix} 9 & 5 & 2 & 4 \\ -5 & -1 & -3 & 5 \\ 6 & -8 & 10 & 3 \end{pmatrix}
\]

\[
C = \begin{pmatrix} 2 & -1 & 9 \\ 6 & 3 & -5 \\ -2 & 7 & -8 \end{pmatrix}
\]

### Part a: \( AB \) (Matrix Multiplication)

Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix. Here, \( A \) is a \( 3 \times 4 \) matrix and \( B \) is also a \( 3 \times 4 \) matrix. Thus, \( AB \) is **not defined** since the number of columns in \( A \) does not equal the number of rows in \( B \).

### Part b: \( A + B \) (Matrix Addition)

Matrix addition is defined when the matrices have the same dimensions. Both \( A \) and \( B \) are \( 3 \times 4 \) matrices, so we can add them:

\[
A + B = \begin{pmatrix} 2 & 1 & 0 & 3 \\ -1 & 0 & 2 & 4 \\ 4 & -2 & 7 & 0 \end{pmatrix} + \begin{pmatrix} 9 & 5 & 2 & 4 \\ -5 & -1 & -3 & 5 \\ 6 & -8 & 10 & 3 \end{pmatrix}
\]

\[
A + B = \begin{pmatrix} 2+9 & 1+5 & 0+2 & 3+4 \\ -1-5 & 0-1 & 2-3 & 4+5 \\ 4+6 & -2-8 & 7+10 & 0+3 \end{pmatrix} = \begin{pmatrix} 11 & 6 & 2 & 7 \\ -6 & -1 & -1 & 9 \\ 10 & -10 & 17 & 3 \end{pmatrix}
\]

### Part c: \( A - B \) (Matrix Subtraction)

Matrix subtraction is also defined when the matrices have the same dimensions. Therefore, we can subtract \( B \) from \( A \):

\[
A - B = \begin{pmatrix} 2 & 1 & 0 & 3 \\ -1 & 0 & 2 & 4 \\ 4 & -2 & 7 & 0 \end{pmatrix} - \begin{pmatrix} 9 & 5 & 2 & 4 \\ -5 & -1 & -3 & 5 \\ 6 & -8 & 10 & 3 \end{pmatrix}
\]

\[
A - B = \begin{pmatrix} 2-9 & 1-5 & 0-2 & 3-4 \\ -1+5 & 0+1 & 2+3 & 4-5 \\ 4-6 & -2+8 & 7-10 & 0-3 \end{pmatrix} = \begin{pmatrix} -7 & -4 & -2 & -1 \\ 4 & 1 & 5 & -1 \\ -2 & 6 & -3 & -3 \end{pmatrix}
\]

### Part d: \( A + C \) (Matrix Addition)

Matrix \( A \) is \( 3 \times 4 \) and matrix \( C \) is \( 3 \times 3 \), so their dimensions are not the same. Therefore, **\( A + C \) is not defined**.

### Part e: \( B - C \) (Matrix Subtraction)

Similarly, matrix \( B \) is \( 3 \times 4 \) and matrix \( C \) is \( 3 \times 3 \). Since their dimensions are not the same, **\( B - C \) is not defined**.

### Summary of Results:
- \( AB \) is **not defined**.
- \( A + B \) = \( \begin{pmatrix} 11 & 6 & 2 & 7 \\ -6 & -1 & -1 & 9 \\ 10 & -10 & 17 & 3 \end{pmatrix} \)
- \( A - B \) = \( \begin{pmatrix} -7 & -4 & -2 & -1 \\ 4 & 1 & 5 & -1 \\ -2 & 6 & -3 & -3 \end{pmatrix} \)
- \( A + C \) is **not defined**.
- \( B - C \) is **not defined**.

Would you like further details on any specific calculation, or do you have any questions?

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**5 Related Questions:**
1. How do you determine when matrix multiplication is possible?
2. Can you explain why matrix dimensions must match for addition or subtraction?
3. What is the difference between a scalar multiplication and matrix multiplication?
4. How do you find the inverse of a matrix, and when is it possible?
5. What are some real-world applications of matrix operations?

**Tip:** Always check the dimensions of matrices before attempting to perform operations like addition, subtraction, or multiplication to ensure they are valid.

Determine the following for the matrices A, B, and C: a) AB, b) A + B, c) A - B, d) A + C, e) B - C. The given matrices are: A = [3x4 matrix], B = [3x4 matrix], and C = [3x3 matrix].

Solve matrix operations for given 3x4 and 3x3 matrices. Learn how to perform matrix multiplication, addition, and subtraction, including conditions for when these operations are defined.

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