To solve the problem, we need to find the value of \( n \) such that \( P^n = 5I - 8P \), where \( P = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} \) and \( I \) is the identity matrix of order 2.
### Step 1: Calculate \( 5I - 8P \)
First, we calculate \( 5I \) and \( 8P \).
The identity matrix \( I \) of order 2 is:
\[
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]
Now, calculate \( 5I \):
\[
5I = 5 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}
\]
Next, calculate \( 8P \):
\[
8P = 8 \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} = \begin{pmatrix} 16 & -8 \\ 40 & -24 \end{pmatrix}
\]
Now, we find \( 5I - 8P \):
\[
5I - 8P = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} - \begin{pmatrix} 16 & -8 \\ 40 & -24 \end{pmatrix} = \begin{pmatrix} 5 - 16 & 0 - (-8) \\ 0 - 40 & 5 - (-24) \end{pmatrix}
\]
\[
= \begin{pmatrix} -11 & 8 \\ -40 & 29 \end{pmatrix}
\]
### Step 2: Calculate \( P^2 \)
Next, we calculate \( P^2 \):
\[
P^2 = P \cdot P = \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix} \cdot \begin{pmatrix} 2 & -1 \\ 5 & -3 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( 2 \cdot 2 + (-1) \cdot 5 = 4 - 5 = -1 \)
- First row, second column: \( 2 \cdot (-1) + (-1) \cdot (-3) = -2 + 3 = 1 \)
- Second row, first column: \( 5 \cdot 2 + (-3) \cdot 5 = 10 - 15 = -5 \)
- Second row, second column: \( 5 \cdot (-1) + (-3) \cdot (-3) = -5 + 9 = 4 \)
Thus,
\[
P^2 = \begin{pmatrix} -1 & 1 \\ -5 & 4 \end{pmatrix}
\]
### Step 3: Calculate \( P^4 \)
Now, we calculate \( P^4 = P^2 \cdot P^2 \):
\[
P^4 = \begin{pmatrix} -1 & 1 \\ -5 & 4 \end{pmatrix} \cdot \begin{pmatrix} -1 & 1 \\ -5 & 4 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( -1 \cdot (-1) + 1 \cdot (-5) = 1 - 5 = -4 \)
- First row, second column: \( -1 \cdot 1 + 1 \cdot 4 = -1 + 4 = 3 \)
- Second row, first column: \( -5 \cdot (-1) + 4 \cdot (-5) = 5 - 20 = -15 \)
- Second row, second column: \( -5 \cdot 1 + 4 \cdot 4 = -5 + 16 = 11 \)
Thus,
\[
P^4 = \begin{pmatrix} -4 & 3 \\ -15 & 11 \end{pmatrix}
\]
### Step 4: Calculate \( P^6 \)
Next, we calculate \( P^6 = P^4 \cdot P^2 \):
\[
P^6 = \begin{pmatrix} -4 & 3 \\ -15 & 11 \end{pmatrix} \cdot \begin{pmatrix} -1 & 1 \\ -5 & 4 \end{pmatrix}
\]
Calculating the elements:
- First row, first column: \( -4 \cdot (-1) + 3 \cdot (-5) = 4 - 15 = -11 \)
- First row, second column: \( -4 \cdot 1 + 3 \cdot 4 = -4 + 12 = 8 \)
- Second row, first column: \( -15 \cdot (-1) + 11 \cdot (-5) = 15 - 55 = -40 \)
- Second row, second column: \( -15 \cdot 1 + 11 \cdot 4 = -15 + 44 = 29 \)
Thus,
\[
P^6 = \begin{pmatrix} -11 & 8 \\ -40 & 29 \end{pmatrix}
\]
### Step 5: Compare \( P^6 \) with \( 5I - 8P \)
Now, we compare \( P^6 \) with \( 5I - 8P \):
\[
P^6 = \begin{pmatrix} -11 & 8 \\ -40 & 29 \end{pmatrix} \quad \text{and} \quad 5I - 8P = \begin{pmatrix} -11 & 8 \\ -40 & 29 \end{pmatrix}
\]
Since they are equal, we have:
\[
P^6 = 5I - 8P
\]
### Conclusion
Thus, the value of \( n \) is \( 6 \).
