https://file.aiquickdraw.com/mathbot/file/1727927263875yionqu0v.jpg

In the image, the problem asks to perform the indicated matrix operation. Here's how to approach the operation:

### Problem Description:
We have two matrices that are involved in operations:

1. Matrix 1:
\[
\begin{bmatrix}
4 & 1 \\
2 & 5
\end{bmatrix}
\]

2. Matrix 2:
\[
\begin{bmatrix}
4 & 0 & 4
\end{bmatrix}
\]

3. Matrix 3:
\[
\begin{bmatrix}
0 & 2 \\
2 & 2 \\
-4 & 1
\end{bmatrix}
\]

The problem appears to ask for a matrix multiplication operation between these matrices.

### Explanation:
1. **Matrix 2** is a row vector of dimensions \( 1 \times 3 \).
2. **Matrix 3** is a matrix of dimensions \( 3 \times 2 \).

To multiply these two matrices, the number of columns of the first matrix (Matrix 2) must equal the number of rows of the second matrix (Matrix 3). Since the dimensions align (Matrix 2 is \(1 \times 3\) and Matrix 3 is \(3 \times 2\)), the multiplication can be performed.

### Step-by-Step Matrix Multiplication:

Multiply \( \text{Matrix 2} \times \text{Matrix 3} \):
\[
\begin{bmatrix}
4 & 0 & 4
\end{bmatrix}
\times
\begin{bmatrix}
0 & 2 \\
2 & 2 \\
-4 & 1
\end{bmatrix}
=
\begin{bmatrix}
(4 \times 0) + (0 \times 2) + (4 \times -4) & (4 \times 2) + (0 \times 2) + (4 \times 1)
\end{bmatrix}
\]

Performing the calculations:
\[
\begin{bmatrix}
(0 + 0 - 16) & (8 + 0 + 4)
\end{bmatrix}
=
\begin{bmatrix}
-16 & 12
\end{bmatrix}
\]

### Next Steps:
Now, we need to add **Matrix 1** to the resulting matrix from the previous operation. Matrix 1 is \(2 \times 2\), but we need to verify the resulting matrix dimensions for compatibility with addition. However, if the addition cannot happen due to incompatible dimensions, the answer will be **DNE (Does Not Exist)**.

Given that Matrix 1 is \(2 \times 2\) and the resulting matrix from the multiplication is \(1 \times 2\), the dimensions do not match for matrix addition.

### Conclusion:
The answer is **DNE** because the matrices cannot be added due to incompatible dimensions.

---

Would you like any further clarification or details on matrix multiplication or any other topic?

### Related Questions:
1. What are the rules for matrix addition and multiplication?
2. Can you explain why matrix dimensions must align for addition?
3. What are the properties of matrix operations, such as commutative and associative properties?
4. How do you determine if a matrix inverse exists?
5. What is the difference between a square matrix and a rectangular matrix?

### Tip:
When performing matrix operations, always verify that the matrix dimensions are compatible for the intended operation (addition, multiplication, etc.) before proceeding.

This page explains how to solve a matrix operation involving multiplication and addition, with step-by-step guidance. It illustrates the rules for matrix multiplication and the conditions under which matrix addition is possible. Ideal for Grades 10-12 or college-level linear algebra students.

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