please show clear thanks， linear algebra question Problem 9: Prove that an upper-triangular matrix is invertible if and only if all of itsdiagonal entries are nonzero, using the following approach: Let A = (aij) be an upper-triangular matrix of size n × n. Consider the associated linear map TA : F → F.ThenTÃ(еj) = αjjej + Σαijli, j = 1,2,...,n,where e₁,e2, ..., en, is the standard basis of Fn. Suppose each diagonal entry aii #0. Then show, recursively, that e; € im(TÂ), for i 1,2,..., n. Conclude that=rank(TA)n, whence A is invertible. Conversely, suppose that one of the diagonalentries of A is equal to 0. Show that rank A

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Answered: Problem 9: Prove that an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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please show clear thanks， linear algebra question

Problem 9: Prove that an upper-triangular matrix is invertible if and only if all of its
diagonal entries are nonzero, using the following approach: Let A = (aij) be an upper-
triangular matrix of size n × n. Consider the associated linear map TA : F → F.
Then
TÃ(еj) = αjjej + Σαijli, j = 1,2,...,n,
where e₁,e2, ..., en, is the standard basis of Fn. Suppose each diagonal entry aii #
0. Then show, recursively, that e; € im(TÂ), for i 1,2,..., n. Conclude that
=
rank(TA)
n, whence A is invertible. Conversely, suppose that one of the diagonal
entries of A is equal to 0. Show that rank A <n.
-
(Note: By combining problems 4 and 9, we get the theorem that an upper-triangular
matrix is invertible if and only if its determinant is nonzero. Notice that we have not
used the multiplicativity of the determinant in this approach!)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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