Advanced 2014 | JEE 2014 | Discussion Page

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

Option A,B

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

Discussion Forum