Given `X=[(2,0,2),(1,0,-1)]Y=[(3,-1,0),(-2,0,-1)]`, find Z such `X+Y+Z=O`

Answer

To solve the problem, we need to find the matrix \( Z \) such that:

\[ X + Y + Z = O \]

where \( O \) is the null matrix (all elements are zero). We will follow these steps:

### Step 1: Define the matrices \( X \) and \( Y \)

Given:
\[ X = \begin{pmatrix} 2 & 0 & 2 \\ 1 & 0 & -1 \end{pmatrix}, \quad Y = \begin{pmatrix} 3 & -1 & 0 \\ -2 & 0 & -1 \end{pmatrix} \]

### Step 2: Calculate \( X + Y \)

To find \( Z \), we first need to compute \( X + Y \):

\[ X + Y = \begin{pmatrix} 2 & 0 & 2 \\ 1 & 0 & -1 \end{pmatrix} + \begin{pmatrix} 3 & -1 & 0 \\ -2 & 0 & -1 \end{pmatrix} \]

We add the corresponding elements:

\[ X + Y = \begin{pmatrix} 2 + 3 & 0 - 1 & 2 + 0 \\ 1 - 2 & 0 + 0 & -1 - 1 \end{pmatrix} = \begin{pmatrix} 5 & -1 & 2 \\ -1 & 0 & -2 \end{pmatrix} \]

### Step 3: Set up the equation for \( Z \)

Now, we can substitute \( X + Y \) into the equation:

\[ Z = O - (X + Y) \]

Since \( O \) is the null matrix:

\[ O = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

Thus, we have:

\[ Z = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} - \begin{pmatrix} 5 & -1 & 2 \\ -1 & 0 & -2 \end{pmatrix} \]

### Step 4: Calculate \( Z \)

Now we perform the subtraction:

\[ Z = \begin{pmatrix} 0 - 5 & 0 - (-1) & 0 - 2 \\ 0 - (-1) & 0 - 0 & 0 - (-2) \end{pmatrix} = \begin{pmatrix} -5 & 1 & -2 \\ 1 & 0 & 2 \end{pmatrix} \]

### Final Answer

Thus, the matrix \( Z \) is:

\[ Z = \begin{pmatrix} -5 & 1 & -2 \\ 1 & 0 & 2 \end{pmatrix} \]