Let the matrices `A=[{:( sqrt3,-2),(0,1):}]` and P be any orthogonal matrix such that Q = PAP' and let `Rne [r_0] _(2-2)=P'Q^(6) P`then

Options

A. `r_(11) =81 `
B. `r_(1l) =81 sqrt3`
C. ` r_(11) -4sqrt3`
D. ` r_(11)=-sqrt3`

Answer

To solve the problem step by step, we need to find \( R = P' Q^6 P \) where \( Q = PAP' \) and \( A = \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \).

### Step 1: Calculate \( Q \)
Given \( Q = PAP' \), we need to compute \( Q \).

1. **Find \( P' \)**: Since \( P \) is an orthogonal matrix, \( P' \) is its transpose.
2. **Compute \( Q \)**: We need to multiply \( A \) by \( P' \) and then by \( P \).

### Step 2: Calculate \( Q^6 \)
Next, we need to compute \( Q^6 \). This involves multiplying \( Q \) by itself six times. However, we can simplify the computation using properties of matrices.

### Step 3: Substitute \( Q \) into \( R \)
Now we substitute \( Q \) into the equation for \( R \):
\[ R = P' Q^6 P = P' (P A P')^6 P \]

### Step 4: Simplify \( R \)
Using the property of orthogonal matrices, we have:
\[ P' P = I \quad \text{(identity matrix)} \]
Thus, we can simplify \( R \):
\[ R = P' (P A P')^6 P = P' P A^6 P' P = A^6 \]

### Step 5: Compute \( A^6 \)
Now we need to compute \( A^6 \). We can do this by first calculating \( A^2 \), \( A^3 \), \( A^4 \), \( A^5 \), and finally \( A^6 \).

1. **Calculate \( A^2 \)**:
\[ A^2 = A \cdot A = \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \]
Performing the multiplication:
\[ A^2 = \begin{pmatrix} 3 & -2\sqrt{3} \\ 0 & 1 \end{pmatrix} \]

2. **Calculate \( A^3 \)**:
\[ A^3 = A^2 \cdot A = \begin{pmatrix} 3 & -2\sqrt{3} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \]
Performing the multiplication:
\[ A^3 = \begin{pmatrix} 3\sqrt{3} & -6 \\ 0 & 1 \end{pmatrix} \]

3. **Calculate \( A^4 \)**:
\[ A^4 = A^3 \cdot A = \begin{pmatrix} 3\sqrt{3} & -6 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \]
Performing the multiplication:
\[ A^4 = \begin{pmatrix} 9 & -6\sqrt{3} \\ 0 & 1 \end{pmatrix} \]

4. **Calculate \( A^5 \)**:
\[ A^5 = A^4 \cdot A = \begin{pmatrix} 9 & -6\sqrt{3} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \]
Performing the multiplication:
\[ A^5 = \begin{pmatrix} 27 & -18 \\ 0 & 1 \end{pmatrix} \]

5. **Calculate \( A^6 \)**:
\[ A^6 = A^5 \cdot A = \begin{pmatrix} 27 & -18 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} \sqrt{3} & -2 \\ 0 & 1 \end{pmatrix} \]
Performing the multiplication:
\[ A^6 = \begin{pmatrix} 81 & -54 \\ 0 & 1 \end{pmatrix} \]

### Final Result
Thus, we have:
\[ R = A^6 = \begin{pmatrix} 81 & -54 \\ 0 & 1 \end{pmatrix} \]