Add, Subtract and Scalar Multiply Matrices - Free Printable

Problem: Perform the given matrix operations.

#### Part (a):
$$
\begin{bmatrix}
5 & -7 \\
4 & 6
\end{bmatrix}
+
\begin{bmatrix}
-8 & -3 \\
4 & 0
\end{bmatrix}
$$

Solution:
To add two matrices, we add their corresponding elements. The matrices are:
$$
A = \begin{bmatrix}
5 & -7 \\
4 & 6
\end{bmatrix}, \quad
B = \begin{bmatrix}
-8 & -3 \\
4 & 0
\end{bmatrix}
$$

The sum $ A + B $ is calculated as follows:
$$
A + B = \begin{bmatrix}
5 + (-8) & -7 + (-3) \\
4 + 4 & 6 + 0
\end{bmatrix}
$$

Perform the element-wise addition:
- Top-left element: $ 5 + (-8) = -3 $
- Top-right element: $ -7 + (-3) = -10 $
- Bottom-left element: $ 4 + 4 = 8 $
- Bottom-right element: $ 6 + 0 = 6 $

Thus, the resulting matrix is:
$$
A + B = \begin{bmatrix}
-3 & -10 \\
8 & 6
\end{bmatrix}
$$

Final Answer for (a):
$$
\boxed{\begin{bmatrix}
-3 & -10 \\
8 & 6
\end{bmatrix}}
$$

---

#### Part (b):
$$
\begin{bmatrix}
5 & -7 & 5 \\
-2 & 3 & -4
\end{bmatrix}
+
\begin{bmatrix}
-1 & -3 & \\
9 & -3 & \\
10 & -9 &
\end{bmatrix}
$$

Solution:
First, observe that the second matrix is not fully written. It appears to be a $ 3 \times 2 $ matrix, but it should match the dimensions of the first matrix ($ 2 \times 3 $) for addition to be valid. Assuming the missing entries are zeros, the second matrix can be written as:
$$
B = \begin{bmatrix}
-1 & -3 & 0 \\
9 & -3 & 0 \\
10 & -9 & 0
\end{bmatrix}
$$

However, this still does not match the dimensions of the first matrix. For matrix addition, both matrices must have the same dimensions. Since the problem statement seems to have an error in the dimensions, we cannot proceed with the addition as stated.

Note: If the problem intended for the second matrix to be a $ 2 \times 3 $ matrix, please provide the correct entries for the missing elements.

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#### Part (c):
$$
\begin{bmatrix}
x + y & -y & x \\
3x & -9 & -2 \\
-3x & 4 & 4y - x
\end{bmatrix}
-
\begin{bmatrix}
2x & y & 3x \\
-3 & 12 & -5 \\
y & x & 2 + 3y
\end{bmatrix}
$$

Solution:
To subtract two matrices, we subtract their corresponding elements. The matrices are:
$$
A = \begin{bmatrix}
x + y & -y & x \\
3x & -9 & -2 \\
-3x & 4 & 4y - x
\end{bmatrix}, \quad
B = \begin{bmatrix}
2x & y & 3x \\
-3 & 12 & -5 \\
y & x & 2 + 3y
\end{bmatrix}
$$

The difference $ A - B $ is calculated as follows:
$$
A - B = \begin{bmatrix}
(x + y) - 2x & (-y) - y & x - 3x \\
3x - (-3) & -9 - 12 & -2 - (-5) \\
-3x - y & 4 - x & (4y - x) - (2 + 3y)
\end{bmatrix}
$$

Simplify each element:
- Top-left element: $ (x + y) - 2x = x + y - 2x = -x + y = y - x $
- Top-middle element: $ (-y) - y = -y - y = -2y $
- Top-right element: $ x - 3x = -2x $
- Middle-left element: $ 3x - (-3) = 3x + 3 $
- Middle-middle element: $ -9 - 12 = -21 $
- Middle-right element: $ -2 - (-5) = -2 + 5 = 3 $
- Bottom-left element: $ -3x - y = -3x - y $
- Bottom-middle element: $ 4 - x = 4 - x $
- Bottom-right element: $ (4y - x) - (2 + 3y) = 4y - x - 2 - 3y = y - x - 2 $

Thus, the resulting matrix is:
$$
A - B = \begin{bmatrix}
y - x & -2y & -2x \\
3x + 3 & -21 & 3 \\
-3x - y & 4 - x & y - x - 2
\end{bmatrix}
$$

Final Answer for (c):
$$
\boxed{\begin{bmatrix}
y - x & -2y & -2x \\
3x + 3 & -21 & 3 \\
-3x - y & 4 - x & y - x - 2
\end{bmatrix}}
$$

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Summary of Answers:

1. Part (a): $\boxed{\begin{bmatrix} -3 & -10 \\ 8 & 6 \end{bmatrix}}$
2. Part (b): Cannot be solved due to dimension mismatch.
3. Part (c): $\boxed{\begin{bmatrix} y - x & -2y & -2x \\ 3x + 3 & -21 & 3 \\ -3x - y & 4 - x & y - x - 2 \end{bmatrix}}$