Matrix Inverse

Definition

For a square matrix $A$, its inverse (denoted as $A^{-1}$) is a matrix that satisfies: $A{A}^{-1}=A^{-1}A=I$ where $I$ is the identity matrix.

Properties

1. Only square matrices can have inverses.
2. Not all square matrices have inverses. Those that do are called invertible or non-singular.
3. If $A^{-1}$ exists, it is unique.
4. $A$ is invertible if and only if the determinant $\det A\ne 0$
5. $A$ is invertible if and only if $A$ have no zero eigenvalue

Calculation

Adjugate Matrix

See Adjugate Matrix

Polynomials

If $A$ is an invertible matrix, its inverse can sometimes be expressed as a polynomial of $A$. One common method involves the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic equation.

Given the characteristic polynomial of $A$: $p(\lambda )=\det (\lambda I-A)={\lambda }^n+c_{n-1}{\lambda }^{n-1}+\cdots +c_1\lambda +c_0$ By the Cayley-Hamilton theorem, we have: $p(A)=A^n+c_{n-1}{A}^{n-1}+\cdots +c_{1}A+c_{0}I=0$ If $c_0\ne 0$, we can rearrange this equation to solve for $A^{-1}$: $A^n+c_{n-1}{A}^{n-1}+\cdots +c_{1}A+c_{0}I=0$ $A(A^{n-1}+c_{n-1}{A}^{n-2}+\cdots +c_{1}I)=-c_{0}I$ Multiplying both sides by $-\frac{1}{c_0}$, we get: $A^{-1}=-\frac{1}{c_0}(A^{n-1}+c_{n-1}{A}^{n-2}+\cdots +c_{1}I)$ Thus, $A^{-1}$ can be expressed as a polynomial of $A$.

Other Computational Methods

For larger matrices, numerical methods such as:

• Gaussian elimination
• LU decomposition are commonly used to compute the inverse.