## frx 2 years ago Let A be the matrix $A=\left[\begin{matrix}1 & -2\\ -2 &4\end{matrix}\right]$ decide all the 2x2 matrices B such that AB=BA=0, where 0 is the zeromatrix

1. frx

$\left[\begin{matrix}1 & -2 \\ -2 & 4\end{matrix}\right]\left[\begin{matrix}a &b \\ b & c\end{matrix}\right]=\left[\begin{matrix}0 &0 \\ 0& 0\end{matrix}\right]$ $\left[\begin{matrix}a & b \\ c & d\end{matrix}\right]\left[\begin{matrix}1 &-2\\ -2 & 4\end{matrix}\right]=\left[\begin{matrix}0 & 0\\ 0 &0\end{matrix}\right]$ So all I can figure is that for AB=BA=0 B can be$B=A ^{-1}=\left[\begin{matrix}4 & 2 \\ 2 & 1\end{matrix}\right]$

2. frx

But is there any other solution than the inverse?

3. frx

I guess it could be $\left[\begin{matrix}1 & -2 \\ -2 & 4\end{matrix}\right]\left[\begin{matrix}0 & 0 \\ 0& 0\end{matrix}\right]=\left[\begin{matrix}0 & 0 \\ 0& 0\end{matrix}\right]$ $\left[\begin{matrix}0 & 0 \\ 0& 0\end{matrix}\right]\left[\begin{matrix}1 & -2 \\ -2 & 4\end{matrix}\right]=\left[\begin{matrix}0 & 0 \\ 0& 0\end{matrix}\right]$ but so the two solutions for AB=BA=0 should be B=A^-1 and B=0, am I right?

4. Zarkon

$$A$$ is not invertable...so how can $$B=A^{-1}$$

5. frx

It's not I guess, A is not invertable if AX=0 is the definition I remember right now, is that right?

6. Zarkon

$$Det(A)=0$$ so $$A^{-1}$$ does not exist

7. frx

Ok, but is there any other solution than B equals the zerovector?

8. Zarkon

yes..there are infinitly many solutions

9. Zarkon

you have 2 of the solutions

10. frx

So how do I show that it has infinity many solutions?

11. Zarkon

show that $$B=\left[\begin{matrix}4 & 2 \\ 2 & 1\end{matrix}\right]$$ is a solution

12. Zarkon

then $$B\cdot t$$ is a solution for all real numbers $$t$$

13. frx

So B can be the same as what the inverse of A would have been if it existed?

14. Zarkon

it is not really the same

15. Zarkon

remember you have to divide by the det(A) for the shortcut way to find the inverse

16. frx

Oh I get it, it can't be the inverse since the definition says that A^-1*A=I isn't that right?

17. frx

I= Identity matrix

18. Zarkon

yes

19. frx

Thank you for your help! :)

20. Zarkon

np