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Question

Let M and N be two 3 × 3 matrices such that MN = NM. Further, if MN2 and M2=N4, then


Correct Answer
A
determinant of (M2+MN2) is 0.
Correct Answer
B
there is a 3 × 3 non - zero matrix U such that (M2+MN2) U is zero matrix.
Your Answer
C
determinant of (M2+MN2)1
Your Answer
D
for a 3 × 3 matrix U, if (M2+MN2)Uequals the zero matrix, then U is the zero matrix

Solution

The correct options are
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.
PLAN (i) If A and B are two non - zero matrices and AB = BA, then (A - B) (A + B) = A2B2
(ii) The determinant of the product of the matrices is equal to product of their individual determinants, i.e. |AB| = |A||B|.
Given, M2=N4M2N4=0
  (MN2)(M+N2)=0    [as MN = NM]
Also,                 MN2
  M+N2=0  det(M+N2)=0
Also,  det (M2+MN2) = (det M) (det (M + N2))
 = (det M) (0) = 0
As,     det(M2+MN2)=0
Thus, there exists a non - zero matrix U such that
(M2+MN2)U=0
 

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