Inverse Matrix
The inverse of a square matrix A, written A^-1, is the unique matrix such that A times A^-1 equals the identity matrix.
Formula
A A^{-1} = A^{-1} A = I
Definition
The inverse of a matrix $A$ is another matrix $A^{-1}$ such that when multiplied together they give the identity matrix: $A A^{-1} = A^{-1} A = I$, and not every matrix has one, a matrix is invertible (nonsingular) if and only if $\det(A) \neq 0$. For $2\times 2$: if $A = \begin{bmatrix}a & b\\c & d\end{bmatrix}$, then $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix}d & -b\\-c & a\end{bmatrix}$; larger matrices use row reduction or cofactor methods, giving $A^{-1} = \text{adj}(A)/\det(A)$, where $\text{adj}(A)$ is the adjugate (transpose of the cofactor matrix). By the Neumann series, $(I - B)^{-1} = \sum_{k=0}^{\infty} B^k$ whenever $\|B\| < 1$ in operator norm, the matrix generalization of the geometric series $1/(1-r) = \sum r^n$; the set of invertible $n \times n$ matrices forms the general linear group $GL_n(F)$, whose rich subgroup structure ($SL_n$, $O_n$, $U_n$, $Sp_n$) underlies differential geometry, physics, and representation theory.
Example
For $A = \begin{bmatrix}2 & 1\\5 & 3\end{bmatrix}$, $A^{-1} = \begin{bmatrix}3 & -1\\-5 & 2\end{bmatrix}$, and checking, $A A^{-1} = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} = I$. For $A = \begin{bmatrix}4 & 7\\2 & 6\end{bmatrix}$, $\det(A) = 10$, so $A^{-1} = (1/10)\begin{bmatrix}6 & -7\\-2 & 4\end{bmatrix}$. The inverse is used to solve matrix equations: if $Ax = b$, then $x = A^{-1}b$ when $A$ is invertible, the matrix analog of dividing both sides of an equation by a number.
Key Insight
An inverse matrix "undoes" the original transformation, just like inverse functions undo each other: if $A$ scrambles some information, $A^{-1}$ unscrambles it, and only matrices with nonzero determinant have this property. Matrix invertibility is the central concept of linear algebra, and the set of invertible matrices forming the general linear group $GL_n(F)$, with its rich subgroup structure, underlies much of differential geometry, physics, and representation theory.