Jelly, 11 10 bytes
ṖṖ€$ƬÆḊṂ>0
Uses Sylvester's criterion.
How it works
ṖṖ€$ƬÆḊṂ>0 Main link. Argument: M (matrix)
$Ƭ Do the following until a fixed point is encountered.
Ṗ Pop; remove the last row of the matrix.
Ṗ€ Pop each; remove the last entry of each row.
ÆḊ Take the determinants of the resulting minors.
Ṃ Take the minimum.
>0 Test if the least determinant is positive, i.e., if all determinants are.