Inverse Matrix

An inverse matrix is a matrix that, when multiplied with another matrix, returns the Identity Matrix. A matrix is invertible if it has an inverse matrix, and singular if it doesn’t.

Definition §

Let there be a $n\times n$ square matrix $\mathbf{A}$. If there exists a matrix inverse ${\mathbf{A}}^{-1}$ such that:

$$\mathbf{A}{\mathbf{A}}^{-1}={\mathbf{A}}^{-1}\mathbf{A}={\mathbf{I}}_n$$

• ${\mathbf{I}}_n$ is the identity matrix of size $n$

If ${\mathbf{A}}^{-1}$ does not exist, then $\mathbf{A}$ is said to be singular. If ${\mathbf{A}}^{-1}$ exists, $\mathbf{A}$ is invertible

This also leads to some interesting consequences:

• ${\mathbf{A}}^{-1}$ is unique i.e. there can only exist one inverse for a given matrix
• ${\mathbf{A}}^{-1}$ has the same size as $\mathbf{A}$ i.e. same number of rows and columns.
• If $\mathbf{A}$ is invertible, then ${\mathbf{A}}^{-1}$ is invertible and $({\mathbf{A}}^{-1}{)}^{-1}=\mathbf{I}$.
• The inverse of the identity matrix is the identity matrix: ${\mathbf{I}}^{-1}=\mathbf{I}$
• The zero matrix is singular
• A matrix that has a row or column consisting entirely of zeros is singular.

$\mathbf{A}$ is invertible iff it has a non-zero Determinant:

$$\mathbf{A}\text{ is invertible}⟺\det (\mathbf{A})\ne 0$$

$\mathbf{A}$ is invertible iff it can be row-reduced to $\mathbf{I}$.

$$\mathbf{A}\text{ is invertible}⟺\mathbf{A}\sim {\mathbf{I}}_n$$

Properties §

Let $\mathbf{A}$ and $\mathbf{B}$ be $n\times n$ invertible matrices and $k$ be a non-zero scalar, then:

• $(k\mathbf{A}{)}^{-1}=\frac{1}{k}{\mathbf{A}}^{-1}$
• $(\mathbf{AB}{)}^{-1}={\mathbf{B}}^{-1}{\mathbf{A}}^{-1}$
• $({\mathbf{A}}^k{)}^{-1}=({\mathbf{A}}^{-1}{)}^k$ if $k\in \mathbb{Z},k\ge 1$
• $({\mathbf{A}}^T{)}^{-1}=({\mathbf{A}}^{-1}{)}^T$

Obtaining the Inverse Matrix §