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

Options

  1. ADeterminant of M 2 + M N 2 is 0
  2. BThere is a 3 × 3 non - zero matrix U such that M 2 + M N 2 U is the zero matrix
  3. CDeterminant of M 2 + M N 2 ≥ 1
  4. DFor a 3 × 3 matrix U , if M 2 + M N 2 U equals the zero matrix then U is the zero matrix

Correct answer

A. Determinant of M 2 + M N 2 is 0

Step-by-step solution

M 2 = N 4   ⇒     M 2 - N 4 = 0   ⇒ M - N 2   M +   N 2 = 0 (As M, N commute.) Also, M   ≠ N 2 ,   D e t   M - N 2   M + N 2 = 0 As M - N 2 is not null ⇒ D e t   M + N 2 = 0 Also D e t   M 2 + M N 2 = D e t   M   D e t   M + N 2 = 0 ⇒ There exist non - null U such that M 2 + M N 2   U = 0

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