matrix equation
h = (f + g) + 13
2f = h + 7
g = 0.5f
a, $\begin{bmatrix}1 & 1 & 1 \\ 2 & 0 & 1 \\ 0.5 & 1 & 0 \end{bmatrix}\begin{bmatrix}f \\ g \\ h \end{bmatrix}=\begin{bmatrix}13 \\ 7 \\ 0 \end{bmatrix}$
b, $\begin{bmatrix}1 & -1 & -1 \\ 2 & -1 & 0 \\ 1 & -0.5 & 0 \end{bmatrix}\begin{bmatrix}f \\ g \\ h \end{bmatrix}=\begin{bmatrix}13 \\ 7 \\ 0 \end{bmatrix}$
c, $\begin{bmatrix}-1 & -1 & 1 \\ 2 & -1 & 0 \\ -0.5 & 1 & 0 \end{bmatrix}\begin{bmatrix}f \\ g \\ h \end{bmatrix}=\begin{bmatrix}13 \\ 7 \\ 0 \end{bmatrix}$
$\begin{bmatrix}-1 & -1 & 1 \\ 2 & 0 & -1 \\ -0.5 & 1 & 0 \end{bmatrix}\begin{bmatrix}f \\ g \\ h \end{bmatrix}=\begin{bmatrix}13 \\ 7 \\ 0 \end{bmatrix}$

Question

Accepted Answer

To solve the matrix equation problem, we first need to convert the given system of equations into matrix form. Let's start by writing down the equations:

1. $ h = (f + g) + 13 $ can be rewritten as $ -f - g + h = 13 $
2. $ 2f = h + 7 $ can be rewritten as $ 2f - h = 7 $ or $ 2f + 0g - h = 7 $
3. $ g = 0.5f $ can be rewritten as $ -0.5f + g = 0 $ or $ -0.5f + g + 0h = 0 $

Now, we can represent this system of linear equations in the form $ A\mathbf{x} = \mathbf{b} $, where $ A $ is the coefficient matrix, $ \mathbf{x} = \begin{bmatrix} f \ g \ h \end{bmatrix} $, and $ \mathbf{b} = \begin{bmatrix} 13 \ 7 \ 0 \end{bmatrix} $.

### Step 1: Identify the coefficients for each equation
- For the first equation $ -f - g + h = 13 $, the coefficients are $ -1 $ (for $ f $), $ -1 $ (for $ g $), and $ 1 $ (for $ h $).
- For the second equation $ 2f + 0g - h = 7 $, the coefficients are $ 2 $ (for $ f $), $ 0 $ (for $ g $), and $ -1 $ (for $ h $).
- For the third equation $ -0.5f + g + 0h = 0 $, the coefficients are $ -0.5 $ (for $ f $), $ 1 $ (for $ g $), and $ 0 $ (for $ h $).

### Step 2: Form the coefficient matrix $ A $
Using the coefficients from the equations, the matrix $ A $ is:
$$
A = \begin{bmatrix} -1 & -1 & 1 \ 2 & 0 & -1 \ -0.5 & 1 & 0 \end{bmatrix}
$$
And the vector $ \mathbf{b} $ is:
$$
\mathbf{b} = \begin{bmatrix} 13 \ 7 \ 0 \end{bmatrix}
$$
So the matrix equation is:
$$
\begin{bmatrix} -1 & -1 & 1 \ 2 & 0 & -1 \ -0.5 & 1 & 0 \end{bmatrix} \begin{bmatrix} f \ g \ h \end{bmatrix} = \begin{bmatrix} 13 \ 7 \ 0 \end{bmatrix}
$$
This matches the last matrix equation provided in the options.

# Answer:
The correct matrix equation is $\begin{bmatrix} -1 & -1 & 1 \ 2 & 0 & -1 \ -0.5 & 1 & 0 \end{bmatrix} \begin{bmatrix} f \ g \ h \end{bmatrix} = \begin{bmatrix} 13 \ 7 \ 0 \end{bmatrix}$ (the last one in the given options).

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QUESTION IMAGE

Question

Explanation:

Step 1: Identify the coefficients for each equation

Step 2: Form the coefficient matrix \( A \)

Answer: