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

Let M and N be two 3 ×3 matrices such that MN = NM. Further, if M ≠N2 and M2=N4 , then
  1. Determinant of M2+MN2 is 0
  2. There is a 3 ×3 non - zero matrix U such that M2+MN2U is the zero matrix
  3. Determinant of M2+MN2 ≥1
  4. For a 3 ×3 matrix U, if M2+MN2U equals the zero matrix then U is the zero matrix

Solution

M2=N4 ⇒  M2-N4=0 ⇒M-N2 M+ N2=0 (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=0

Asked in: JEE Advanced 2014 (Paper 1)

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