Inverse matrix

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


\mathbf{A}^{-1}\mathbf{A} = \mathbf{I} :=
\begin{pmatrix}
1      & 0 & 0 & \cdots & 0 \\
0      & 1 & 0 & \cdots & 0 \\
0      & 0 & 1 & \cdots & 0 \\
\cdots &   &   &        &   \\ 
0      & 0 & 0 & \cdots & 1 \\
\end{pmatrix}.

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

If for an invertible matrix A it holds that


\mathbf{A}\mathbf{B} = \mathbf{I}

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


\mathbf{A}^{-1}\mathbf{A}\mathbf{B} = \mathbf{A}^{-1}\mathbf{I} = \mathbf{A}^{-1}
\quad\Longrightarrow\quad \mathbf{B}= \mathbf{A}^{-1}.

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.

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