Let m and N be two 3x3 matrices such that MN=NM. Further if `M!=N^2` and `M^2=N^4` then which of the following are correct.

Options

A. determinant of `(M^(2)+Mn^(2))` is 0
B. there is a `3xx3` non-zero matrix U such that `(M^(2)+MN^(2))U` is the zero matrix
C. determinant of `(M^(2)+MN^(2)) ge 1`
D. for a `3xx3` matrix `U`, is the zero matrix

Answer

A, B

Explanation

`M^(2)=N^(4)`
`implies M^(2)-N^(4)=O`
`implies (M-N^(2)) (M+N^(2))=O" "` (as M, N commute)
Also, `M ne N^(2)`.
det. `((M-N^(2))(M+N^(2)))=0`
as `M ne N^(2) implies` dte. `(M+N^(2))=0`
Also, det `(M^(2)+MN^(2))=("det. M") ("det. "(M+N^(2)))=0`
Hence, there exist non-null U such that `(M^(2)+MN^(2)) U=O`.