Let A and B be n ´ n and m ´ m matrices, respectively. The matrix
where each O denotes a matrix of zeros, is called a block-diagonal matrix with two diagonal blocks A and B. [ Note that the two O’s in eq(a) do not have the same dimensions if ]. An example is
For any square matrix A and B,
Let A be n ´ n and B be m ´ m. Let be the n ´ n unit matrix. Then
so that by the product formula Theorem 5.11, we have
can be regarded as a function of the n rows of A. It is then obvious that
it satisfies Axioms 1 and 2 of a determinant function.