Adjoint or Adjugate
The adjoint of A, ADJ(A) is the transpose of the matrix formed by taking the cofactor of each element of A.
ADJ(A) A = det(A) I
If det(A) != 0, then A-1 = ADJ(A) / det(A) but this is a numerically and computationally poor way of calculating the inverse.
A matrix, A, is fully decomposable (or reducible) if there exists a permutation matrix P such that PTAP is of the form [B C; 0 D] where B and D are square.
A matrix, A, is partly-decomposable if there exist permutation matrices P and Q such that PTAQ is of the form [B C; 0 D] where B and D are square.
A matrix that is not even partly-decomposable is fully-indecomposable.
A matrix is singular if it has no inverse.
A matrix A is singular iff det(A)=0.
A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix over a continuous uniform distribution on its entries, it will almost surely not be singular.