If ` [(1,2,1)][(1,2,0),(2,0,1),(1,0,2)] [(0),(2),(x)]` = 0 , then the value of x is

Options

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

Answer

To solve the equation given by the matrix multiplication, we need to find the value of \( x \) such that:

\[ \begin{pmatrix} 1 & 2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{pmatrix} \begin{pmatrix} 0 \\ 2 \\ x \end{pmatrix} = 0 \]

### Step 1: Multiply the first two matrices

First, we will multiply the first matrix \( \begin{pmatrix} 1 & 2 & 1 \end{pmatrix} \) with the second matrix \( \begin{pmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{pmatrix} \).

Calculating the first row of the product:

- The first element:
\[ 1 \cdot 1 + 2 \cdot 2 + 1 \cdot 1 = 1 + 4 + 1 = 6 \]

- The second element:
\[ 1 \cdot 2 + 2 \cdot 0 + 1 \cdot 0 = 2 + 0 + 0 = 2 \]

- The third element:
\[ 1 \cdot 0 + 2 \cdot 1 + 1 \cdot 2 = 0 + 2 + 2 = 4 \]

So, the result of the multiplication is:
\[ \begin{pmatrix} 6 & 2 & 4 \end{pmatrix} \]

### Step 2: Multiply the result with the third matrix

Next, we multiply the resulting matrix \( \begin{pmatrix} 6 & 2 & 4 \end{pmatrix} \) with the third matrix \( \begin{pmatrix} 0 \\ 2 \\ x \end{pmatrix} \).

Calculating the final result:

- The result is:
\[ 6 \cdot 0 + 2 \cdot 2 + 4 \cdot x = 0 + 4 + 4x = 4 + 4x \]

### Step 3: Set the result equal to zero

Since we know that the entire multiplication equals the zero matrix, we set the equation:
\[ 4 + 4x = 0 \]

### Step 4: Solve for \( x \)

Now, we can solve for \( x \):
\[ 4x = -4 \]
\[ x = -1 \]

Thus, the value of \( x \) is:
\[ \boxed{-1} \]