If `[[x], [y], [z]]=(1)/(40)[[5, 10, -5], [-5, -2, 13], [10, -4, 6]][[5], [0], [5]]`, then `x+y+z=`

Options

A. `3`
B. `0`
C. `2`
D. `1`

Answer

To solve the problem, we need to find the values of \(x\), \(y\), and \(z\) from the given matrix equation:

\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{40} \begin{bmatrix} 5 & 10 & -5 \\ -5 & -2 & 13 \\ 10 & -4 & 6 \end{bmatrix} \begin{bmatrix} 5 \\ 0 \\ 5 \end{bmatrix} \]

### Step 1: Calculate the matrix product

First, we will multiply the two matrices on the right-hand side.

Let:
\[ A = \begin{bmatrix} 5 & 10 & -5 \\ -5 & -2 & 13 \\ 10 & -4 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 0 \\ 5 \end{bmatrix} \]

To find \(AB\), we calculate each element of the resulting matrix:

1. First row:
\[ 5 \cdot 5 + 10 \cdot 0 + (-5) \cdot 5 = 25 + 0 - 25 = 0 \]

2. Second row:
\[ -5 \cdot 5 + (-2) \cdot 0 + 13 \cdot 5 = -25 + 0 + 65 = 40 \]

3. Third row:
\[ 10 \cdot 5 + (-4) \cdot 0 + 6 \cdot 5 = 50 + 0 + 30 = 80 \]

Thus, the product \(AB\) is:
\[ AB = \begin{bmatrix} 0 \\ 40 \\ 80 \end{bmatrix} \]

### Step 2: Multiply by \(\frac{1}{40}\)

Now we multiply the resulting matrix by \(\frac{1}{40}\):
\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{40} \begin{bmatrix} 0 \\ 40 \\ 80 \end{bmatrix} = \begin{bmatrix} \frac{0}{40} \\ \frac{40}{40} \\ \frac{80}{40} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} \]

From this, we can identify:
\[ x = 0, \quad y = 1, \quad z = 2 \]

### Step 3: Calculate \(x + y + z\)

Now, we can find \(x + y + z\):
\[ x + y + z = 0 + 1 + 2 = 3 \]

### Final Answer

Thus, the final answer is:
\[ \boxed{3} \]