JEE Advanced | 2014MatricesHard
Question
Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M ≠ N and M2 = N2, then
Options
A.Determinant of (M2 + MN2) is 0
B.There is a 3 × 3 non-zero matrix U such that (M2 + MN2)U is the zero matrix
C.Determinant of (M2 + MN2) ≥ 1
D.For a 3 × 3 matrix U, if (M2 + MN2)U equals the zero matrix then U is the zero matrix
Solution
M ≠ N2 ⇒ M - N2 ≠ 0
M2 - N4 = 0 ⇒ (M - N2)(M + N2) + N2M - MN2 = 0
as MN = NM
⇒ MN2 = NMN
= NNM
= N2M
So, (M - N2)(M + N2) = 0
So either M + N2 = 0 or M - N2 and M + N2 both are singular.
So, there exist a 3 × 3 non-zero matrix U i.e., M - N2 such that
(M + N2)U = 0 ⇒ (M2 + MN2)U = 0
Also, |M2 + MN2| = |M| |M + N2| = 0
M2 - N4 = 0 ⇒ (M - N2)(M + N2) + N2M - MN2 = 0
as MN = NM
⇒ MN2 = NMN
= NNM
= N2M
So, (M - N2)(M + N2) = 0
So either M + N2 = 0 or M - N2 and M + N2 both are singular.
So, there exist a 3 × 3 non-zero matrix U i.e., M - N2 such that
(M + N2)U = 0 ⇒ (M2 + MN2)U = 0
Also, |M2 + MN2| = |M| |M + N2| = 0
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