Linear and multilinear algebra; matrix theory Proposition

Solving a linear system with an inverse matrix

Builds on Invertible matrix System of linear equations

Statement

Let $A$ be an invertible matrix with inverse $B = A^{-1}$ . Then for each vector $b$ of the right size, the system of linear equations $Ax = b$ 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 $x = A^{-1}Ax = A^{-1}b$ .

Reading path

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Invertible matrix concept
Linear equation definition
System of linear equations definition
Solving a linear system with an inverse matrix statement

Tag	`3RQX3T70NX`
URI	`m2://3RQX3T70NX`
Kind	statement, proposition
Writing	W0 stub
MSC	15A09
Status	This copy of the site has no build data. Run `m2 status 3RQX3T70NX`.
About	Invertible matrix
Uses	Invertible matrix, System of linear equations
Changed	4 November 2026

Solving a linear system with an inverse matrix

Statement

Remarks

Reading path

Further reading

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