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Let A ∈ Mn×n(F ) be an upper triangular matrix.
(a) Show that A is invertible ⇐⇒ every diagonal entry of A is nonzero. (Hint
for ⇒: Recall that A is invertible iff rank(A) = n. First show that a11 = 0.
So we can use the first column and elementary column operations to make
a12 = · · · = a1n = 0. Then use Homework 11 Textbook Sec. 3.2 Exercise 11
and mathematical induction.)
(b) Show that when A is invertible, its inverse matrix is also upper triangular.
Sagot :
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