If `x=[(3,1,1),(5,2,3)]` and `y=[(2,1,1),(7,2,4)]`, what is matrix z, such that x+y+z=0.

Options

A. `[(-5,-2,-2),(-12,-4,-7)]`
B. `[(-2,-1,-1),(-7,-2,-4)]`
C. `[(-3,-1,-1),(-5,-2,-3)]`
D. None of these

Answer

To find the matrix \( z \) such that \( x + y + z = 0 \), we can rearrange the equation to solve for \( z \):

\[ z = - (x + y) \]

### Step 1: Define the matrices
Let:
\[ x = \begin{pmatrix} 3 & 1 & 1 \\ 5 & 2 & 3 \end{pmatrix}, \quad y = \begin{pmatrix} 2 & 1 & 1 \\ 7 & 2 & 4 \end{pmatrix} \]

### Step 2: Add the matrices \( x \) and \( y \)
We will add the corresponding elements of matrices \( x \) and \( y \):

\[ x + y = \begin{pmatrix} 3 + 2 & 1 + 1 & 1 + 1 \\ 5 + 7 & 2 + 2 & 3 + 4 \end{pmatrix} \]

Calculating each element:
- First row:
- First element: \( 3 + 2 = 5 \)
- Second element: \( 1 + 1 = 2 \)
- Third element: \( 1 + 1 = 2 \)

- Second row:
- First element: \( 5 + 7 = 12 \)
- Second element: \( 2 + 2 = 4 \)
- Third element: \( 3 + 4 = 7 \)

Thus, we have:
\[ x + y = \begin{pmatrix} 5 & 2 & 2 \\ 12 & 4 & 7 \end{pmatrix} \]

### Step 3: Find matrix \( z \)
Now, we substitute \( x + y \) into the equation for \( z \):
\[ z = - (x + y) = - \begin{pmatrix} 5 & 2 & 2 \\ 12 & 4 & 7 \end{pmatrix} \]

This means we will take the negative of each element:
\[ z = \begin{pmatrix} -5 & -2 & -2 \\ -12 & -4 & -7 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( z \) is:
\[ z = \begin{pmatrix} -5 & -2 & -2 \\ -12 & -4 & -7 \end{pmatrix} \]

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