   Chapter 1.6, Problem 27E

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# A square matrix A = [ a i   j ] n with a i   j = 0 for all i > j is called upper triangular. Prove or disprove each of the following statements.The set of all upper triangular matrices is closed with respect to matrix addition in M n ( ℝ ) .The set of all upper triangular matrices is closed with respect to matrix multiplication in M n ( ℝ ) .If A and B are square and the product A B is upper triangular, then at least one of A or B is upper triangular.

(a)

To determine

Whether the set of all upper triangular matrices is closed with respect to matrix addition in Mn().

Explanation

Given information:

A square matrix A=[aij]n with aij=0 for all i>j is called upper triangular.

Formula used:

Addition in Mm×n() is defined by

[aij]m×n+[bij]m×n=[cij]m×n, where cij=aij+bij.

Calculation:

Let A=[aij]n with aij=0 for all i

(b)

To determine

Whether the set of all upper triangular matrices is closed with respect to matrix multiplication in Mn().

(c)

To determine

If A and B are squares and the product AB is upper triangular, then at least one of A or B is upper triangular.

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