If `[(x),(y),(z)]=(1)/(17) [(1,5,1),(8,6,-9),(10,-1,-7)][(8),(1),(4)]` then the value of xyz is

Options

A. 0
B. 5
C. 8
D. 6

Answer

To solve the problem, we need to perform matrix multiplication and then extract the values of \(x\), \(y\), and \(z\) from the resulting matrix. Finally, we will calculate the product \(xyz\).

### Step-by-step Solution:

1. **Set Up the Matrices**:
We are given:
\[ \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \frac{1}{17} \begin{bmatrix} 1 & 5 & 1 \\ 8 & 6 & -9 \\ 10 & -1 & -7 \end{bmatrix} \begin{bmatrix} 8 \\ 1 \\ 4 \end{bmatrix} \]

2. **Perform Matrix Multiplication**:
We will multiply the 3x3 matrix by the 3x1 column matrix. The resulting matrix will be a 3x1 matrix.

- For the first row:
\[ 1 \cdot 8 + 5 \cdot 1 + 1 \cdot 4 = 8 + 5 + 4 = 17 \]

- For the second row:
\[ 8 \cdot 8 + 6 \cdot 1 + (-9) \cdot 4 = 64 + 6 - 36 = 34 \]

- For the third row:
\[ 10 \cdot 8 + (-1) \cdot 1 + (-7) \cdot 4 = 80 - 1 - 28 = 51 \]

So, the result of the multiplication is:
\[ \begin{bmatrix} 17 \\ 34 \\ 51 \end{bmatrix} \]

3. **Multiply by \(\frac{1}{17}\)**:
Now we multiply the resulting matrix by \(\frac{1}{17}\):
\[ \begin{bmatrix} \frac{17}{17} \\ \frac{34}{17} \\ \frac{51}{17} \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \]

4. **Extract Values of \(x\), \(y\), and \(z\)**:
From the resulting matrix, we can see:
\[ x = 1, \quad y = 2, \quad z = 3 \]

5. **Calculate \(xyz\)**:
Now, we calculate the product:
\[ xyz = 1 \cdot 2 \cdot 3 = 6 \]

### Final Answer:
The value of \(xyz\) is \(6\).

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