Let M And N Be Two 3 × 3 Matrices Such That MN = NM. Further, If M ...

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4. Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N 2 and M 2= N 4, then

Q. Let M and N be two 3×3 matrices such that MN=NM. Further, if M=N2 and M2=N4, then 2924 252 JEE AdvancedJEE Advanced 2014Matrices Report Error A

determinant of (M2+MN2) is 0

33% B

there is a 3×3 non-zero matrix U such that (M2+MN2) U is zero matrix

37% C

determinant of (M2+MN2)≥1

15% D

for a 3×3 matrix U, if (M2+MN2) U equals the zero matrix, then U is the zero matrix

15%

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