Use the definition of Ax to write the matrix equation as a vector equation.
Start 4 By 2 Table 1st Row 1st Column 4 2nd Column 1 2nd Row 1st Column negative 6 2nd Column 8 3rd Row 1st Column 9 2nd Column 9 4st Row 1st Column 0 2nd Column 8 EndTable Start 2 By 1 Table 1st Row 1st Column 4 2nd Row 1st Column 3 EndTable equals Start 4 By 1 Table 1st Row 1st Column 19 2nd Row 1st Column 0 3rd Row 1st Column 63 4st Row 1st Column 24 EndTable 
  
  
 
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Part 1
The matrix equation written as a vector equation is 
  
enter your response here.

To express the given matrix equation as a vector equation, we'll start with the general form of the matrix equation \( A \mathbf{x} = \mathbf{b} \), where:

- \( A \) is a \( 4 \times 2 \) matrix,
- \( \mathbf{x} \) is a \( 2 \times 1 \) column vector,
- \( \mathbf{b} \) is a \( 4 \times 1 \) column vector.

### Matrix Equation:
\[
\begin{pmatrix}
4 & 1 \\
-6 & 8 \\
9 & 9 \\
0 & 8
\end{pmatrix}
\begin{pmatrix}
4 \\
3
\end{pmatrix}
=
\begin{pmatrix}
19 \\
0 \\
63 \\
24
\end{pmatrix}
\]

### Now, breaking this into a vector equation:
Each row of the matrix \( A \) multiplied by the vector \( \mathbf{x} \) results in the corresponding element of the column vector \( \mathbf{b} \).

\[
4(4) + 1(3) = 19
\]
\[
-6(4) + 8(3) = 0
\]
\[
9(4) + 9(3) = 63
\]
\[
0(4) + 8(3) = 24
\]

Thus, the matrix equation written as a vector equation is:

\[
4(4) + 1(3) = 19
\]
\[
-6(4) + 8(3) = 0
\]
\[
9(4) + 9(3) = 63
\]
\[
0(4) + 8(3) = 24
\]

This is the vector equation equivalent to the given matrix equation.

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Would you like further clarification or any additional details?

### Related Questions:
1. What are the properties of matrix multiplication in vector equations?
2. How do you solve for \( \mathbf{x} \) in the equation \( A \mathbf{x} = \mathbf{b} \)?
3. How does the size of the matrix and vector affect the solution?
4. Can we generalize the vector equation for larger matrices?
5. What is the geometric interpretation of solving a matrix-vector equation?

**Tip:** When multiplying matrices, ensure the number of columns in the first matrix equals the number of rows in the second for proper multiplication.

Use the definition of Ax to write the matrix equation as a vector equation. Start 4 By 2 Table 1st Row 1st Column 4 2nd Column 1 2nd Row 1st Column negative 6 2nd Column 8 3rd Row 1st Column 9 2nd Column 9 4st Row 1st Column 0 2nd Column 8 EndTable Start 2 By 1 Table 1st Row 1st Column 4 2nd Row 1st Column 3 EndTable equals Start 4 By 1 Table 1st Row 1st Column 19 2nd Row 1st Column 0 3rd Row 1st Column 63 4st Row 1st Column 24 EndTable.

Learn how to express a 4x2 matrix equation as a vector equation. This problem breaks down the steps of multiplying a matrix by a vector and solving the system of linear equations. Suitable for high school or college students studying linear algebra.

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