Let `M` and `N` be two `3xx3` 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` if `(M^(2)+MN^(2))U` equals the zero
matrix, then `U` is the zero matrix

Answer

A, B

Explanation

Given, `MN = NM, M ne N^(2) and M^(2) = N^(4) `
Then, `M^(2) = N^(2)` lt brgt `rArr (M+N^(2)) (M-N^2) = 0`
`therefore M+N^(2) = 0 " " [because M ne N^(2)]`
`rArr abs( M+ N^(2)) = 0`
(a) `abs( M^(2)+ MN^(2)) = abs(M) abs( M+ N^(2)) = 0`
`therefore` Option (a) is correct.
`( M^(2)+ MN^(2))U = M( M+ N^(2))U = 0`
`therefore` Option (b) is correct.
(c) `becauseabs( M^(2)+ MN^(2)) =0` from opttion (a)
`therefore abs(M^(2) +MN^(2))cancel(ge) 1`
`therefore` OPtion (c ) is incorrect.
(d)If ` AX = 0 and abs(A) = 0` then X can be non-zero.