Question
Mathematics Question on Matrices
Let M and N be two 3×3 matrices such that MN=NM. Further, if M=N2 and M2=N4, then
A
determinant of (M2+MN2) is 0
B
there is a 3×3 non-zero matrix U such that (M2+MN2) U is zero matrix
C
determinant of (M2+MN2)≥1
D
for a 3×3 matrix U, if (M2+MN2) U equals the zero matrix, then U is the zero matrix
Answer
there is a 3×3 non-zero matrix U such that (M2+MN2) U is zero matrix
Explanation
Solution
M2=N4
⇒M2−N4=O
⇒(M−N2)(M+N2)=O
As M, N commute.
Also, M=N2,Det((M−N2)(M+N2))=0
As M−N2 is not null
⇒Det(M+N2)=0
Also Det (M2+MN2)=( Det M)( Det (M+N2))=0
⇒ There exist non-null U such that (M2+MN2)U=O