Inverse Matrix
Functions & Advanced AlgebraThe 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 you multiply them together, you get the identity matrix. Not every matrix has an inverse.
Example
For $A = \begin{bmatrix}2 & 1\\5 & 3\end{bmatrix}$, $A^{-1} = \begin{bmatrix}3 & -1\\-5 & 2\end{bmatrix}$. Check: $\begin{bmatrix}2 & 1\\5 & 3\end{bmatrix} \times \begin{bmatrix}3 & -1\\-5 & 2\end{bmatrix} = \begin{bmatrix}6-5 & -2+2\\15-15 & -5+6\end{bmatrix} = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix} = I$.
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. Only matrices with nonzero determinant have inverses.
Definition
For a square matrix $A$, its inverse $A^{-1}$ satisfies $A A^{-1} = A^{-1} A = I$. A matrix is invertible (nonsingular) iff $\det(A) \neq 0$. 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.
Example
$A = \begin{bmatrix}4 & 7\\2 & 6\end{bmatrix}$. $\det(A) = 24 - 14 = 10$. $A^{-1} = (1/10)\begin{bmatrix}6 & -7\\-2 & 4\end{bmatrix} = \begin{bmatrix}0.6 & -0.7\\-0.2 & 0.4\end{bmatrix}$. Verify: $A A^{-1} = I_2$.
Key Insight
The inverse is used to solve matrix equations: if $Ax = b$, then $x = A^{-1}b$ (when $A$ is invertible). This is the matrix analog of dividing both sides of an equation by a number.
Definition
A square matrix $A$ is invertible iff $\det(A) \neq 0$, equivalently iff $A$ has full rank, equivalently iff $0$ is not an eigenvalue. The inverse $A^{-1} = \text{adj}(A)/\det(A)$, 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).
Example
Neumann series: $(I - B)^{-1} = I + B + B^2 + \ldots$ is an infinite series of matrices converging whenever $\|B\| < 1$. This is the matrix generalization of the geometric series $1/(1-r) = \sum r^n$.
Key Insight
Matrix invertibility is the central concept of linear algebra. 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.