If `A` and `B` are different matrices satisfying `A^(3) = B^(3)` and
`A^(2) B = B^(2) A`, then

Options

A. `det (A^(2) + B^(2))` must be zero
B. det `(A-B)` must be zero
C. `det (A^(2) + B^(2))` as well as `det (A - B)` must be zero
D. alteast one of `det (A^(2) + B^(2))` or `det (A - B)` must be zero

Answer

D

Explanation

`because A^(3) - A^(2) B = B^(3) - B^(2) A `
`rArr A^(2) (A-B) = B^(2)(B-A)`
or `(A^(2) + B^(2)) (A-B) =0`
or det `(A^(2)+B^(2)) cdot det (A-B) = 0`
Either det `(A^(2) + B^(2)) = 0` or det `(A-B) = 0`