Inverse of a matrix: statement, true or false?

HelpNeeder

New member
Joined
Feb 17, 2021
Messages
30
If A is an n x n matrix and Ax = ej is consistent for every j ∊ {1, 2, ..., n}, then A is invertible.

Could someone please help me understand why this statement is true/false?
 

lex

Junior Member
Joined
Mar 3, 2021
Messages
236
It is true. It implies that Ax=b has a solution for all b in \(\displaystyle \mathbb{R^n}\),
which means A has a pivot in every row, which means A can be transformed into \(\displaystyle \text{I}_n\) using e.r.o. s (since A is n x n, the pivots must be on the diagonal), therefore A is invertible.

In more detail:
1617876191671.png
Now (2) implies that A has a pivot in every row (by your previous work on Theorem 4):
1617876268002.png
A is n x n, so the pivots are on the diagonals, therefore it can be transformed into \(\displaystyle \text{I}_n\) by elementary row operations.
Therefore A is invertible.
1617876394914.png
 
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