Inverse and Transpose

(Inverse)

Consider a square matrix $\boldsymbol{A} \in \mathbb{R}^{n\times n}$. Let matrix $\boldsymbol{B} \in \mathbb{R}^{n\times n}$ have the property that $\boldsymbol{A}\boldsymbol{B} = \boldsymbol{I}_{n} = \boldsymbol{B}\boldsymbol{A}$. $\boldsymbol{B}$ is called the inverse of $\boldsymbol{A}$ and is denoted by $\boldsymbol{A}^{-1}$.

If the inverse of a matrix does not exist, its called singular/non-invertible. Otherwise, it’s called regular/invertible/nonsingular.

(Inverse of a $2 \times 2$-matrix)

Consider a matrix

$$\begin{align} \boldsymbol{A} \coloneqq \begin{bmatrix} a_{11}&a_{12} \\ a_{21}&a_{22} \end{bmatrix} \in \mathbb{R}^{2\times2}. \end{align}$$

if we multiply $\boldsymbol{A}$ with

$$\begin{align} \boldsymbol{A^′} \coloneqq \begin{bmatrix} a_{22}&-a_{12} \\ -a_{21}&a_{11} \end{bmatrix} \end{align}$$

we obtain

$$\begin{align} \boldsymbol{A}\boldsymbol{A^′} \coloneqq \begin{bmatrix} a_{11}a_{22}-a_{12}a_{21}&0 \\ 0&a_{11}a_{22}-a_{12}a_{21} \end{bmatrix} = (a_{11}a_{22}-a_{12}a_{21})\boldsymbol{I}. \end{align}$$

Therefore,

$$\begin{align} \boldsymbol{A^{-1}} \coloneqq \frac{1}{(a_{11}a_{22}-a_{12}a_{21})}\begin{bmatrix} a_{11}a_{22}-a_{12}a_{21}&0 \\ 0&a_{11}a_{22}-a_{12}a_{21} \end{bmatrix} \end{align}$$

if and only if $(a_{11}a_{22}-a_{12}a_{21})\neq 0$.

(Transpose)

For $\boldsymbol{A} \in \mathbb{R}^{m\times n}$, the matrix $\boldsymbol{B} \in \mathbb{R}^{n\times n}$ with $b_{ij}=a_{ji}$ is called the transpose of $\boldsymbol{A}$. We write $\boldsymbol{B} = \boldsymbol{A^T}$.

Properties of Inverses and Transpose

$$\begin{align} \boldsymbol{A}\boldsymbol{A^{-1}} = \boldsymbol{I} = \boldsymbol{A^{-1}}\boldsymbol{A} \tag{2.26} \\ (\boldsymbol{A}\boldsymbol{B})^{-1} = \boldsymbol{B^{-1}}\boldsymbol{A^{-1}} \tag{2.27} \\ (\boldsymbol{A}+\boldsymbol{B})^{-1} \neq \boldsymbol{A^{-1}}+\boldsymbol{B^{-1}} \tag{2.28} \\ (\boldsymbol{A^T})^T=\boldsymbol{A} \tag{2.29} \\ (\boldsymbol{A}\boldsymbol{B})^T=\boldsymbol{B}^T\boldsymbol{A^T} \tag{2.30} \\ (\boldsymbol{A}+\boldsymbol{A})^T=\boldsymbol{A}^T+\boldsymbol{B}^T \tag{2.31} \end{align}$$

(Symmetric Matrix). A square matrix is symmetric iff $\boldsymbol{A}=\boldsymbol{A}^T$. If $\boldsymbol{A}$ is invertible, so is $\boldsymbol{A}^T$.

Connections

• Forward Links: Gaussian Elimination
• Backward Links: Matrix