Let I = `[{:(1, 0, 0),(0 ,1, 0),(0,0,1):}] and P = [{:(1, 0, 0),(0 ,-1, 0),(0,0,-2):}] ` . Then the matrix `p^(3) + 2P^(2)` is equal to

Options

A. P
B. I-P
C. 2I + P
D. 2I - P

Answer

To solve the problem, we need to find \( P^3 + 2P^2 \) for the given matrix \( P \).

### Step 1: Define the matrices
We have:
\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
and
\[ P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \]

### Step 2: Calculate \( P^2 \)
To find \( P^2 \), we multiply \( P \) by itself:
\[ P^2 = P \cdot P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \]

Calculating the product:
\[ P^2 = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot (-1) + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot 0 + 0 \cdot (-2) \\ 0 \cdot 1 + (-1) \cdot 0 + 0 \cdot 0 & 0 \cdot 0 + (-1) \cdot (-1) + 0 \cdot 0 & 0 \cdot 0 + (-1) \cdot 0 + 0 \cdot (-2) \\ 0 \cdot 1 + 0 \cdot 0 + (-2) \cdot 0 & 0 \cdot 0 + 0 \cdot (-1) + (-2) \cdot 0 & 0 \cdot 0 + 0 \cdot 0 + (-2) \cdot (-2) \end{pmatrix} \]
\[ = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

### Step 3: Calculate \( P^3 \)
Next, we calculate \( P^3 \) by multiplying \( P^2 \) by \( P \):
\[ P^3 = P^2 \cdot P = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2 \end{pmatrix} \]

Calculating the product:
\[ P^3 = \begin{pmatrix} 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot (-1) + 0 \cdot 0 & 0 \cdot 1 + 0 \cdot 0 + 4 \cdot (-2) \\ 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0 & 0 \cdot 0 + 1 \cdot (-1) + 0 \cdot 0 & 0 \cdot 0 + 1 \cdot 0 + 0 \cdot (-2) \\ 0 \cdot 1 + 0 \cdot 0 + 4 \cdot 0 & 0 \cdot 0 + 0 \cdot (-1) + 4 \cdot 0 & 0 \cdot 0 + 0 \cdot 0 + 4 \cdot (-2) \end{pmatrix} \]
\[ = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -8 \end{pmatrix} \]

### Step 4: Calculate \( 2P^2 \)
Now we calculate \( 2P^2 \):
\[ 2P^2 = 2 \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 4 \end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8 \end{pmatrix} \]

### Step 5: Calculate \( P^3 + 2P^2 \)
Finally, we add \( P^3 \) and \( 2P^2 \):
\[ P^3 + 2P^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -8 \end{pmatrix} + \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 8 \end{pmatrix} \]
\[ = \begin{pmatrix} 1 + 2 & 0 & 0 \\ 0 & -1 + 2 & 0 \\ 0 & 0 & -8 + 8 \end{pmatrix} = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]

### Final Answer
Thus, the matrix \( P^3 + 2P^2 \) is:
\[ \begin{pmatrix} 3 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]