Question

# 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)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) = A2−B2 (ii) The determinant of the product of the matrices is equal to product of their individual determinants, i.e. |AB| = |A||B|. Given, M2=N4⇒M2−N4=0 ⇒  (M−N2)(M+N2)=0    [as MN = NM] Also,                 M≠N2 ⇒  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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