If P = `[{: (2 , -2, -4),(-1, 3, 4),(1 , -2, -3):}]`, then `P^(5)` equals

Options

A. P
B. 2P
C. `-P`
D. `-2P`

Answer

To find \( P^5 \) for the matrix

\[ P = \begin{pmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{pmatrix} \]

we can use the characteristic polynomial of the matrix \( P \). The characteristic polynomial can be derived from the determinant of \( P - \lambda I \), where \( I \) is the identity matrix.

### Step 1: Find the characteristic polynomial

1. **Set up the matrix \( P - \lambda I \)**:
\[ P - \lambda I = \begin{pmatrix} 2 - \lambda & -2 & -4 \\ -1 & 3 - \lambda & 4 \\ 1 & -2 & -3 - \lambda \end{pmatrix} \]

2. **Calculate the determinant**:
\[ \text{det}(P - \lambda I) = (2 - \lambda) \begin{vmatrix} 3 - \lambda & 4 \\ -2 & -3 - \lambda \end{vmatrix} + 2 \begin{vmatrix} -1 & 4 \\ 1 & -3 - \lambda \end{vmatrix} - 4 \begin{vmatrix} -1 & 3 - \lambda \\ 1 & -2 \end{vmatrix} \]

3. **Calculate the 2x2 determinants**:
\[ \begin{vmatrix} 3 - \lambda & 4 \\ -2 & -3 - \lambda \end{vmatrix} = (3 - \lambda)(-3 - \lambda) - (-2)(4) = \lambda^2 - 9 + 8 = \lambda^2 - 1 \]
\[ \begin{vmatrix} -1 & 4 \\ 1 & -3 - \lambda \end{vmatrix} = -1(-3 - \lambda) - 4(1) = 3 + \lambda - 4 = \lambda - 1 \]
\[ \begin{vmatrix} -1 & 3 - \lambda \\ 1 & -2 \end{vmatrix} = -1(-2) - (3 - \lambda)(1) = 2 - (3 - \lambda) = \lambda - 1 \]

4. **Combine the results**:
\[ \text{det}(P - \lambda I) = (2 - \lambda)(\lambda^2 - 1) + 2(\lambda - 1) - 4(\lambda - 1) \]
\[ = (2 - \lambda)(\lambda^2 - 1) + 2(\lambda - 1) - 4(\lambda - 1) \]
\[ = (2 - \lambda)(\lambda^2 - 1) - 2(\lambda - 1) \]

5. **Expand and simplify**:
\[ = (2\lambda^2 - 2 - \lambda^3 + \lambda) - 2\lambda + 2 = -\lambda^3 + 2\lambda^2 - \lambda + 0 \]

Thus, the characteristic polynomial is:
\[ -\lambda^3 + 2\lambda^2 - \lambda = 0 \]

### Step 2: Use the characteristic polynomial to express \( P^3 \)

From the characteristic polynomial, we can express \( P^3 \) in terms of \( P^2 \) and \( P \):
\[ P^3 = 2P^2 - P \]

### Step 3: Calculate \( P^4 \) and \( P^5 \)

1. **Calculate \( P^4 \)**:
\[ P^4 = P \cdot P^3 = P(2P^2 - P) = 2P^3 - P^2 \]

2. **Calculate \( P^5 \)**:
\[ P^5 = P \cdot P^4 = P(2P^3 - P^2) = 2P^4 - P^3 \]

### Final Result

Thus, we can express \( P^5 \) in terms of lower powers of \( P \) using the characteristic polynomial. The exact matrix calculation can be done by substituting the values of \( P \) into these equations.