DeterminantMedium
Question
Let P and Q be 3 × 3 matrices with P ≠ Q.If P3 = Q3 and P2Q = Q2P, then determinant of(P2 + Q2) is equal to :
Options
A.-1
B.-2
C.1
D.0
Solution
P2Q = Q2P
PQ(P - Q) = 0
∴ PQ = 0 ........(1) (∴P ≠ Q)
and P3 - Q3 = 0
(P - Q) (P2 + PQ + Q2) = 0
∴ P2 + PQ + Q2 = 0 (∴P ≠ Q)
P2 + Q2 = 0 using (1)
so |P2 + Q2| = 0
PQ(P - Q) = 0
∴ PQ = 0 ........(1) (∴P ≠ Q)
and P3 - Q3 = 0
(P - Q) (P2 + PQ + Q2) = 0
∴ P2 + PQ + Q2 = 0 (∴P ≠ Q)
P2 + Q2 = 0 using (1)
so |P2 + Q2| = 0
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