Problem: Under what conditions on their entries are matrices A and B invertible?

Matrix A =

[a b c

d e 0

f 0 0]

Matrix B =

[a b 0

c d 0

0 0 e]

My work so far:

In A: c, e, and f must not equal 0

In B: e must be nonzero, and rows 1 and 2 must not be scalar multiples of one another

Am I missing any other obvious conditions?