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• The structure matrices for most finite Boolean-type lattices and complementation algebras are presented. • Using the structure matrices, the homomorphisms and isomorphisms of BTAs are investigated. • The decomposition of a BTA is considered, which means decomposing a BTA into a product of two BTAs, and a straightforward verifiable necessary and sufficient condition is obtained. This paper provides a systematic matrix description for finite Boolean- type algebras (BTAs). A BTA is decomposed into Boolean-type lattice (BTL) and a complementation algebra (CA). When the object set is finite, the matrix expressions of BTL and CA (and then of BTA) are presented. Via their matrix expression, the construction and certain properties of BTAs are investigated, including the homomorphism and isomorphism, as well as the decomposition. Moreover, straightforward verifiable conditions are obtained to detect the properties above by means of logical matrices operations. [ABSTRACT FROM AUTHOR]