A square matrix P satisfies P2=I-P, where I is identity matrix. If Pn=5 P-31, then n is

Question

A square matrix P satisfies P2=I-P, where I is identity matrix. If Pn=5 P-31, then n is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

∵ P 3= P 2 ⋅ P =( I - P ) P = P - P 2 = P -( I - P )=2 P - I and P 4= P ⋅ P 3= P (2 P - I )=2 P 2- P =2( I - P )- P =2 I-3 P and P 5= P 4 ⋅ P =(2 I -3 P ) P =2 P -3 P 2 =2 P-3(I-P)=5 P-3 I ∴ n=5

Q. A square matrix $P$ satisfies $P^{2} = I - P$ , where $I$ is identity matrix. If $P^{n} = 5 P - 31$ , then $n$ is

$∵ P^{3} = P^{2} \cdot P = (I - P) P = P - P^{2}$
$= P - (I - P) = 2 P - I$
and $P^{4} = P \cdot P^{3} = P (2 P - I) = 2 P^{2} - P$
$= 2 (I - P) - P$
$= 2 I - 3 P$
and $P^{5} = P^{4} \cdot P = (2 I - 3 P) P = 2 P - 3 P^{2}$
$= 2 P - 3 (I - P) = 5 P - 3 I$
$∴ n = 5$

Q. A square matrix P satisfies P2=I−P, where I is identity matrix. If Pn=5P−31, then n is

∵P3=P2⋅P=(I−P)P=P−P2 =P−(I−P)=2P−I and P4=P⋅P3=P(2P−I)=2P2−P =2(I−P)−P =2I−3P and P5=P4⋅P=(2I−3P)P=2P−3P2 =2P−3(I−P)=5P−3I ∴n=5

Q. A square matrix $P$ satisfies $P^{2} = I - P$ , where $I$ is identity matrix. If $P^{n} = 5 P - 31$ , then $n$ is

$∵ P^{3} = P^{2} \cdot P = (I - P) P = P - P^{2}$
$= P - (I - P) = 2 P - I$
and $P^{4} = P \cdot P^{3} = P (2 P - I) = 2 P^{2} - P$
$= 2 (I - P) - P$
$= 2 I - 3 P$
and $P^{5} = P^{4} \cdot P = (2 I - 3 P) P = 2 P - 3 P^{2}$
$= 2 P - 3 (I - P) = 5 P - 3 I$
$∴ n = 5$