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Question

Mathematics Question on Matrices

Let MM and NN be two 3×33 \times 3 matrices such that MN=NMMN = NM. Further, if MN2M \neq N ^{2} and M2=N4M ^{2}= N ^{4}, then

A

determinant of (M2+MN2)(M^2 + MN^2) is 00

B

there is a 3×33 \times 3 non-zero matrix U such that (M2+MN2)(M^2 + MN^2) U is zero matrix

C

determinant of (M2+MN2)1(M^2 + MN^2) \ge 1

D

for a 3×33 \times 3 matrix U, if (M2+MN2)(M^2 + MN^2) U equals the zero matrix, then U is the zero matrix

Answer

there is a 3×33 \times 3 non-zero matrix U such that (M2+MN2)(M^2 + MN^2) U is zero matrix

Explanation

Solution

M2=N4M^{2}=N^{4}
M2N4=O\Rightarrow M^{2}-N^{4}=O
(MN2)(M+N2)=O\Rightarrow\left(M-N^{2}\right)\left(M+N^{2}\right)=O
As M, N commute.
Also, MN2,Det((MN2)(M+N2))=0M \neq N^{2}, \text{Det}\left(\left(M-N^{2}\right)\left(M+N^{2}\right)\right)=0
As MN2M - N ^{2} is not null
Det(M+N2)=0\Rightarrow \text{Det}\left( M + N ^{2}\right)=0
Also Det (M2+MN2)=(\left(M^{2}+M N^{2}\right)=( Det M)(M)\left(\right. Det (M+N2))=0\left.\left(M+N^{2}\right)\right)=0
\Rightarrow There exist non-null UU such that (M2+MN2)U=O\left( M ^{2}+ MN ^{2}\right) U = O