Invertible Matrix

linear-algebra

Definition

Invertible Matrix

A square matrix $A\in F^{m\times n}$ is called invertible if there exists a matrix $A^{-1}\in F^{m\times n}$ such that:

$$\exists A^{-1}\in F^{m\times n}:A\cdot A^{-1}=A^{-1}\cdot A=I_n$$

The matrix $A^{-1}$ is called the inverse of $A$. Not invertible matrices are called singular matrices.

Properties

Let $A,B\in F^{m\times n}$ be two invertible matrices, then:

$$\begin{array}{rl}(A\cdot B{)}^{-1}& =B^{-1}\cdot A^{-1}\\ (A^T{)}^{-1}& =(A^{-1}{)}^T\end{array}$$

Invertibility

A square matrix $A\in F^{m\times n}$ is invertible if and only if its columns (or rows) are linearly independent. Thus, the columns (or rows) should form a basis of $F^n$.

Determinant

Let $A\in F^{n\times n}$ and $\hat{A}=(A_{ij})\in F^{n\times n}$ be the cofactors of a matrix $A$.

$$A^{-1}=\frac{1}{\det A}\cdot {\hat{A}}^T={\left(\frac{A_{ji}}{\det A}\right)}_{1\le i,j\le n}$$