Discussion Forum



1)

Let M and N  be two 3 x3  matrices such that MN=NM  . further , If   $M\neq N^{2}$  and $M^{2}= N^{4}$   then


A) determinant of ($M^{2}+M N^{2}$ ) IS 0

B) there is a 3 x 3 non zero matrix U such that ($M^{2}+M N^{2}$ ) U is zero matrix

C) determination of( $M^{2}+M N^{2}) $ $\geq 1$

D) for a 3x3 matrix U. if ( $M^{2}+M N^{2}) $ U equal the zero matrix, then U is the zero matrix

Answer:

Option A,B

Explanation:

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

   $\Rightarrow$           $ M^{2}-N^{4}=0$

 $\Rightarrow$       $(M-N^{2})(M+N^{2})=0$

                                                         (as MN=NM)

 Also,    $M\neq N^{2}$

  $\Rightarrow$       M+N2=0

$\Rightarrow$     Det (M+N2 )=0

 Also,                 Det (M+MN2 )

   =  (Det M) ((Det (M+N2)

    = (Det M)  (0)= 0

As,  Det (M2+MN2)=0

Thus, there exist non-zero matrix  U such that (M2+MN2)U=0