A = [
2    7
-8   -2
]
B = [
-6   9
-4   0
]
C= [
-4     4
-7    -3
]
AB + AC =  [
?    ?
?    ?
]

We are tasked with calculating \( AB + AC \), where \( A \), \( B \), and \( C \) are matrices. This requires two matrix multiplications followed by matrix addition.

Given:
\[
A = \begin{bmatrix} 2 & 7 \\ -8 & -2 \end{bmatrix}, 
B = \begin{bmatrix} -6 & 9 \\ -4 & 0 \end{bmatrix}, 
C = \begin{bmatrix} -4 & 4 \\ -7 & -3 \end{bmatrix}
\]

### Step 1: Calculate \( AB \)

To multiply \( A \) by \( B \), we follow the standard matrix multiplication rules:

\[
AB = \begin{bmatrix}
2 \cdot (-6) + 7 \cdot (-4) & 2 \cdot 9 + 7 \cdot 0 \\
-8 \cdot (-6) + (-2) \cdot (-4) & -8 \cdot 9 + (-2) \cdot 0
\end{bmatrix}
\]

This gives:
\[
AB = \begin{bmatrix}
-12 - 28 & 18 + 0 \\
48 + 8 & -72 + 0
\end{bmatrix}
= \begin{bmatrix}
-40 & 18 \\
56 & -72
\end{bmatrix}
\]

### Step 2: Calculate \( AC \)

Similarly, multiply \( A \) by \( C \):

\[
AC = \begin{bmatrix}
2 \cdot (-4) + 7 \cdot (-7) & 2 \cdot 4 + 7 \cdot (-3) \\
-8 \cdot (-4) + (-2) \cdot (-7) & -8 \cdot 4 + (-2) \cdot (-3)
\end{bmatrix}
\]

This gives:
\[
AC = \begin{bmatrix}
-8 - 49 & 8 - 21 \\
32 + 14 & -32 + 6
\end{bmatrix}
= \begin{bmatrix}
-57 & -13 \\
46 & -26
\end{bmatrix}
\]

### Step 3: Calculate \( AB + AC \)

Now, add the results of \( AB \) and \( AC \):

\[
AB + AC = \begin{bmatrix}
-40 & 18 \\
56 & -72
\end{bmatrix}
+ \begin{bmatrix}
-57 & -13 \\
46 & -26
\end{bmatrix}
\]

This gives:
\[
AB + AC = \begin{bmatrix}
-40 + (-57) & 18 + (-13) \\
56 + 46 & -72 + (-26)
\end{bmatrix}
= \begin{bmatrix}
-97 & 5 \\
102 & -98
\end{bmatrix}
\]

Thus, \( AB + AC = \begin{bmatrix} -97 & 5 \\ 102 & -98 \end{bmatrix} \).

Would you like further details on any of the steps or clarifications?

### Related Questions:
1. How do you multiply 3x3 matrices?
2. What are the properties of matrix multiplication?
3. How do you find the inverse of a 2x2 matrix?
4. Can matrix multiplication be commutative in certain conditions?
5. What is the determinant of a 2x2 matrix?

### Tip:
When multiplying matrices, always ensure that the number of columns in the first matrix matches the number of rows in the second matrix for the multiplication to be valid.

A = [2  7; -8  -2], B = [-6  9; -4  0], C = [-4  4; -7  -3], AB + AC = ?

Learn how to solve the matrix equation AB + AC for the given matrices A, B, and C. This solution includes detailed steps for matrix multiplication and matrix addition, ideal for students studying linear algebra at the undergraduate or advanced high school level.

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