  Exploded layered tropical (ELT) algebra is an extension of tropical algebra
with a structure of layers. These layers allow us to use classical algebraic
results in order to easily prove analogous tropical results. Specifically we
study the connection between the ELT determinant and linear dependency, and use
a generalized version of Kapranov Theorem proved in [7] (called the Fundamental
Theorem).
  In this paper we prove that an ELT matrix is singular if and only if its rows
are linearly dependent and that the row rank and submatrix rank of an ELT
matrix are equal. We also define an ELT rank for a tropical matrix, and prove
that it is equal to its Kapranov rank. In addition, we formalize the concept of
ELT inner products, and prove ELT versions of some known theorems such as
Cauchy-Schwarz inequality.
