Question

Let M and N be two 3 $\times$ 3 matrices such that MN = NM. Further, if $M\ne N^{2}and{M}^2=N^4$, then

• A. determinant of $(M^2+M{N}^2)$ is 0.
• B. there is a 3 $\times$ 3 non - zero matrix U such that $(M^2+M{N}^2)$ U is zero matrix.
• C. determinant of $(M^2+M{N}^2)\ge 1$
• D. for a 3 $\times$ 3 matrix U, if $(M^2+M{N}^2)$Uequals the zero matrix, then U is the zero matrix

Answer 1

Answer: (A), (B)

The correct options are
A determinant of $(M^2+M{N}^2)$ is 0.
B there is a 3 $\times$ 3 non - zero matrix U such that $(M^2+M{N}^2)$ U is zero matrix.

PLAN (i) If A and B are two non - zero matrices and AB = BA, then (A - B) (A + B) = $A^2-B^2$
(ii) The determinant of the product of the matrices is equal to product of their individual determinants, i.e. |AB| = |A||B|.
Given, $M^2=N^4\Rightarrow M^2-N^4=0$
$\Rightarrow (M-N^2)(M+N^2)=0$ [as MN = NM]
Also, $M\ne N^2$
$\Rightarrow M+N^2=0\Rightarrow det(M+N^2)=0$
Also, det $(M^2+M{N}^2)$ = (det M) (det (M + $N^2$))
= (det M) (0) = 0
As, $det(M^2+M{N}^2)=0$
Thus, there exists a non - zero matrix U such that
$(M^2+M{N}^2)U=0$