Question

Let U be a matrix where u_ij = 0 if i > j, and L be a matrix where l_ij = 0 if i < j.

(a) U is called an upper triangular matrix and L is a lower tri-angular matrix. Explain why.
(b) Prove or disprove: The sum of two upper triangular matrices is an upper triangular matrix.
(c) Prove or disprove: The product of two upper triangular matrices is an upper triangular matrix.

Answer 1

A) U is called an upper triangular matrix because all entries below the principal diagonal element are zeros ( 0 ) since Uij = 0 if i >j also

L is a lower triangular matrix because all entries above the principal diagonal element are zero ( 0 )

B) sum of two upper triangular matrices = upper triangular matrix.

C) product of two upper triangular matrices = upper triangular matrix

Explanation:

A) U is called an upper triangular matrix because all entries below the principal diagonal element are zeros ( 0 ) since Uij = 0 if i >j also

L is a lower triangular matrix because all entries above the principal diagonal element are zero ( 0 ) since Lij = 0 if i < j

B) To prove that sum of two upper triangular matrices

attached below

C) Prove or disprove that product of two upper triangular matrices is an upper triangular matrix

attached below