Inverse Matrix

Definition

Let

• $A,B$ : Square matrix
• $I$ : Identity matrix

If $B$ is such that $AB=BA=I$

Then

• $A$ is called invertible (or nonsingular)
• $B$ is called inverse of $A$

If no such matrix $B$ is found, then $A$ is said to be singular

If $BA=I$, then we say $B$ is left inverse of $A$

If $AB=I$, then we say is right inverse of $A$

Remark

If $A$ is an invertible and $B$ is inverse of $A$, then it is also true that $B$ is invertible and $A$ is inverse of $B$. Thus, when $AB=BA=I$, then we say $A$ and $B$ are inverses of one another.