Question

Let P and Q be `3xx3` matrices with `P!=Q` . If `P^3=""Q^3a n d""P^2Q""=""Q^2P` , then determinant of `(P^2+""Q^2)` is equal to (1) `2` (2) 1 (3) 0 (4) `1`
A. `-2`
B. 1
C. 0
D. `-1`

Answer 1

Correct Answer - C
We have `P^(3)=Q^(3)` and `P^(2)Q=PQ^(2)`
Subtracting, we get
`P^(3)-P^(2)Q=Q^(3)-Q^(2)P`
`P^(2) (P-Q)+Q^(2) (P-Q)=O`
`(P^(2)+Q^(2)) (P-Q)=O`
If `|P^(2)+Q^(2)|ne O` then `P^(2)+Q^(2)` is invertible
`implies P-Q=O` (contradiction)
Hence `|P^(2)+Q^(2)|=0`