# Inverse matrix

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In mathematics, a square matrix **A** may have a **left-inverse matrix** **A**^{−1} defined by

If **A**^{−1} exists, the matrix **A** is called *regular*, *non-singular*, or *invertible*.

If for an invertible matrix **A** it holds that

then the matrix **B** is the **right-inverse** of **A**.
Assume that **A** is invertible and multiply the last equation by the left-inverse

It follows that for any finite-dimensional matrix **A** the right-inverse matrix is equal to the left-inverse matrix, simply called *the* inverse of **A** and indicated by **A**^{−1}.

A necessary and sufficient condition for the inverse **A**^{−1} to exist is the non-vanishing of the determinant: det(**A**) ≠ 0.

See this article for an explicit expression for the elements of **A**^{−1}.