Theorem.

1. Every square matrix is a sum of an upper triangular matrix and a lower triangular matrix.
2. The product of two upper (lower) triangular matrices is an upper (lower) triangular matrix.
3. The transpose of an upper triangular matrix is a low triangular matrix.

Proof. We shall prove only the first of these statements leaving the other statements as exercises. Take any square matrix A of order n. Let B be the matrix such that B(i,j)=A(i,j) if i is greater than j and B(i,j)=0 otherwise (i,j=1,2,...,n). Let C be A-B. Then B is an upper triangular matrix by definition. The matrix C is lower triangular because for every i and j if i is greater than j then

			  C(i,j)=A(i,j)-B(i,j)=A(i,j)-A(i,j)=0