- Home
- Standard 12
- Mathematics
Let $M$ and $N$ be two $3 \times 3$ matrices such that $M N=N M$. Further, if $M \neq N^2$ and $M^2=N^4$, then
$(A)$ determinant of $\left( M ^2+ MN ^2\right)$ is $0$
$(B)$ there is a $3 \times 3$ non-zero matrix $U$ such that $\left( M ^2+ MN ^2\right) U$ is the zero matrix
$(C)$ determinant of $\left( M ^2+ MN ^2\right) \geq 1$
$(D)$ for a $3 \times 3$ matrix $U$, if $\left( M ^2+ MN ^2\right) U$ equals the zero matrix then $U$ is the zero matrix
$(B,D)$
$(B,C)$
$(A,B)$
$(A,D)$
Solution
$M N=N M \& M^2-N^4=0$
$Image$
$(A)$$\left| M ^2+ MN ^2\right|=| M |\left| M + N ^2\right| $
=0 $(A)$ is correct
$(B)$ If $| A |=0$ then $AU =0$ will have $\infty$ solution. Thus $\left( M ^2+ MN ^2\right) U =0$ will have many ' $U$ ' $( B )$ is correct
$(C)$ Obvious wrong.
$(D)$ If $A X=0 \&|A|=0$ then $X$ can be non zero. $(D)$ is wrong
