# Thread: Some basic matrix proofs

1. ## Some basic matrix proofs

Can anyone help me solve these? I'm totally stuck...

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?

2. $\displaystyle \left( A^{-1} \right)^{t}\cdot A^{t}=\left( A\cdot A^{-1} \right)^{t}=I_{n}.$

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

3. $\displaystyle AB = 0$

$\displaystyle A^{-1} AB = A^{-1} 0 = 0$

$\displaystyle (A^{-1}A)B = 0$

$\displaystyle B = 0$

I have no idea why it's making everything lowercase.

EDIT: I guess it was an issue with Firefox.

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

5. 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.