Inverse vs Invertible In linear algebra, they talk about a matrix being an inverse and a matrix being invertible. Is there a difference because they seem like the same thing, but when I reading about them, sometimes they seem like there related, but different. So is their a difference? Or can the words inverse and invertible be used interchangeably?
 A: As said in the comments, inverse is a noun and invertible is an adjective. If a matrix is invertible, then it has an inverse. Here's the definition of an inverse: 
Definition A matrix $B$ is said to be the inverse of a matrix $A$ if and only if $$AB = BA = I,$$ where $I$ is the identity matrix. In this case, we write $B = A^{-1}$. When the matrix $B = A^{-1}$ exists, we say that $A$ is invertible.
Is this clear enough? I will be happy to clarify anything in the comments.
A: Inverse and Invertible does not mean the same.
Matrix $A_{n*n}$ is Invertible when is non-singular or regular, this is:
$\det(A) \neq 0$ and $rank(A) = n$
This means that each column of $A$ is not a linear combination of the rest, so A has full-rank and non-zero determinant, therefore it's regular or non-singular and is invertible as a consequence.
Obviously a matrix has inverse when it is invertible. So if the previous properties don't hold then the matrix $A$ doesn't have an inverse.
Inverse matrices satisfy above conditions and $A\cdot A^{-1} = A^{-1}\cdot A = I$ but this is only true when $A$ is a square matrix.
For non-square matrices we may have right or left inverses see inverse element for more info about these.
