---
tag: 3RQX3T70NX
kind: statement
role: proposition
name: Solving a linear system with an inverse matrix
msc: [15A09]
uses: [0EB9N0S33C, 0JTRMBQ1SA]
about: [0EB9N0S33C]
sources:
  - url: https://github.com/davidaustinm/ula/blob/b6ea263ef23703072b7007485bb7a2d89c5f399b/src/section3-1.xml#L525
    locator: src/section3-1.xml:525
---

## Statement

Let $A$ be an [[invertible matrix|0EB9N0S33C]] with inverse $B = A^{-1}$.
Then for each vector $b$ of the right size, the [[system of linear equations|0JTRMBQ1SA]] $Ax = b$ is consistent.
The vector $x = Bb$ is a solution of it.
So the solution of $Ax = b$ is given by $x = A^{-1}b$.

## Remarks

The check is direct: $A(Bb) = (AB)b = Ib = b$.
This is why an inverse, once known, solves every system with the same matrix $A$.
Uniqueness also holds, since $Ax = b$ gives $x = A^{-1}Ax = A^{-1}b$.