I've encountered a problem in which a 2x2 matrix was given to me. It has the form
\begin{bmatrix} A & B \\ & C \end{bmatrix}
in which the lower left entry is blank. First off, does this even make sense? I don't know if this is standard or just the notation the individual used whom provided it to me. If it is standard, does the determinant exist? If so, is it simply AC-B i.e. treat the blank as $1$, or would I treat the blank as $0$ and the determinant is AC? Any thoughts or ideas? Again, I don't know what to make of the blank entry. In the end, I'm trying to determine the eigenvalues and I don't know if I should include the off diagonals in any way.
Blank entries in matrices, in the absence of other markings, mean zero. This is a nice, if informal, way to save writing or typing time when dealing with larger matrices with many zeros, like diagonal and triangular matrices.
Determinants of sparse matrices, those with many zeros, simplify considerably because many of the terms in the Leibniz expansion of the determinant then evaluate to zero. Thus the determinant of this matrix is simply $AC$.