Q100 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics
$\text { A square matrix } P \text { satisfies } P^2=I-P \text { where } I \text { is identity matrix. If } P^n=5 I-8 P \text {, then } n \text { is equal to }$
- A4
- B7
- C5
- Correct answerD6
Solution
Given $P^2 = I - P$.
We can write $P^2 = -P + I$.
Multiplying by $P$, $P^3 = -P^2 + P = -(I - P) + P = -I + P + P = 2P - I$.
Multiplying by $P$ again, $P^4 = 2P^2 - P = 2(I - P) - P = 2I - 3P$.
Multiplying by $P$ again, $P^5 = 2P - 3P^2 = 2P - 3(I - P) = 2P - 3I + 3P = 5P - 3I$.
Multiplying by $P$ again, $P^6 = 5P^2 - 3P = 5(I - P) - 3P = 5I - 5P - 3P = 5I - 8P$.
Comparing $P^6 = 5I - 8P$ with the given equation $P^n = 5I - 8P$, we find $n = 6$.
Answer: 6
We can write $P^2 = -P + I$.
Multiplying by $P$, $P^3 = -P^2 + P = -(I - P) + P = -I + P + P = 2P - I$.
Multiplying by $P$ again, $P^4 = 2P^2 - P = 2(I - P) - P = 2I - 3P$.
Multiplying by $P$ again, $P^5 = 2P - 3P^2 = 2P - 3(I - P) = 2P - 3I + 3P = 5P - 3I$.
Multiplying by $P$ again, $P^6 = 5P^2 - 3P = 5(I - P) - 3P = 5I - 5P - 3P = 5I - 8P$.
Comparing $P^6 = 5I - 8P$ with the given equation $P^n = 5I - 8P$, we find $n = 6$.
Answer: 6
Hence, the correct answer is option: 'D'.