
Some basic matrix proofs
Can anyone help me solve these? I'm totally stuck...(Thinking)
1. Let A be invertible. Prove that the transpose of A (denoted $\displaystyle A^t$)
is invertible and $\displaystyle (A^t)^{1}=(A^{1})^t$
2. Prove that if A is invertible and $\displaystyle AB=0$, then $\displaystyle B=0$.
Any help on these? (Thinking)

$\displaystyle \left( A^{1} \right)^{t}\cdot A^{t}=\left( A\cdot A^{1} \right)^{t}=I_{n}.$
Thus $\displaystyle \left( A^{1} \right)^{t}\cdot A^{t}\cdot \left( A^{t} \right)^{1}=I_{n}\cdot \left( A^{t} \right)^{1}\implies \left( A^{1} \right)^{t}=\left( A^{t} \right)^{1}.\quad\blacksquare$

$\displaystyle AB = 0 $
$\displaystyle A^{1} AB = A^{1} 0 = 0 $
$\displaystyle (A^{1}A)B = 0 $
$\displaystyle B = 0 $
I have no idea why it's making everything lowercase.
EDIT: I guess it was an issue with Firefox.

You could also use the fact that $\displaystyle \det(A)=\det(A^t)$.

as far as paupsers knows, since these are basic proofs then the use of determinants wouldn't be valid, though he(she) may clarify this.