\[ \pmb{\textit{O}} =\left( \begin{array}{ccccc} 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & \cdots & 0 & 0 \end{array} \right) \]

\[ \pmb{\textit{I}} =\left( \begin{array}{ccccc} 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 1 & 0\\ 0 & 0 & \cdots & 0 & 1 \end{array} \right) \]

次の行列を考える。

\[ \pmb{\textit{A}}=\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) ,\quad \pmb{\textit{B}}=\left( \begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \end{array} \right) \]

\[ \pmb{\textit{x}}=\left( \begin{array}{c} x_{1}\\ x_{2} \end{array} \right) ,\quad \pmb{\textit{y}}=\left( \begin{array}{c} y_{1}\\ y_{2} \end{array} \right) \]

\[ \pmb{\textit{A}}^{T}=\left( \begin{array}{cc} a_{11} & a_{21}\\ a_{12} & a_{22} \end{array} \right) \]

\[ \pmb{\textit{x}}^{T}=\left( \begin{array}{cc} x_{1} & x_{2} \end{array} \right) \]

\[ \pmb{\textit{A}}+\pmb{\textit{B}} =\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) +\left( \begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \end{array} \right) =\left( \begin{array}{cc} a_{11}+b_{11} & a_{12}+b_{12}\\ a_{21}+b_{21} & a_{22}+b_{22} \end{array} \right) \]

\[ \pmb{\textit{x}}+\pmb{\textit{y}} =\left( \begin{array}{c} x_{1}\\ x_{2} \end{array} \right) +\left( \begin{array}{c} y_{1}\\ y_{2} \end{array} \right) =\left( \begin{array}{c} x_{1}+y_{1}\\ x_{2}+y_{2} \end{array} \right) \]

$\lambda$ はスカラー（行列ではない）の定数とする。

\[ \lambda\pmb{\textit{A}} =\lambda\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) =\left( \begin{array}{cc} \lambda a_{11} & \lambda a_{12}\\ \lambda a_{21} & \lambda a_{22} \end{array} \right) \]

\[ \lambda\pmb{\textit{x}} =\lambda\left( \begin{array}{cc} x_{1}\\ x_{2} \end{array} \right) =\left( \begin{array}{cc} \lambda x_{1}\\ \lambda x_{2} \end{array} \right) \]

\[ \pmb{\textit{A}}\pmb{\textit{B}} =\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) \left( \begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \end{array} \right) =\left( \begin{array}{cc} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22}\\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{array} \right) \]

\[ \pmb{\textit{B}}\pmb{\textit{A}} =\left( \begin{array}{cc} b_{11} & b_{12}\\ b_{21} & b_{22} \end{array} \right) \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) =\left( \begin{array}{cc} b_{11}a_{11}+b_{12}a_{21} & b_{11}a_{12}+b_{12}a_{22}\\ b_{21}a_{11}+b_{22}a_{21} & b_{21}a_{12}+b_{22}a_{22} \end{array} \right) \]

\[ \pmb{\textit{A}}\pmb{\textit{x}} =\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) \left( \begin{array}{c} x_{1}\\ x_{2} \end{array} \right) =\left( \begin{array}{c} a_{11}x_{1}+a_{12}x_{2}\\ a_{21}x_{1}+a_{22}x_{2} \end{array} \right) \]

\[ \pmb{\textit{x}}^{T}\pmb{\textit{A}} =\left( \begin{array}{cc} x_{1} & x_{2} \end{array} \right) \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) =\left( \begin{array}{cc} a_{11}x_{1}+a_{21}x_{2} & a_{12}x_{1}+a_{22}x_{2} \end{array} \right) \]

\[ \pmb{\textit{x}}^{T}\pmb{\textit{y}} =\left( \begin{array}{cc} x_{1} & x_{2} \end{array} \right) \left( \begin{array}{c} y_{1}\\ y_{2} \end{array} \right) =x_{1}y_{1}+x_{2}y_{2} \]

次の行列を考える。 \[ \pmb{\textit{A}}=\left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) \]

\[ \pmb{\textit{A}}^{-1} = \frac{1}{a_{11}a_{22}-a_{12}a_{21}} \left( \begin{array}{cc} a_{22} & -a_{12}\\ -a_{21} & a_{11} \end{array} \right) \]

$\pmb{\textit{A}}^{-1}$ は $\pmb{\textit{A}}$ の逆行列という。 ただし、$a_{11}a_{22}-a_{12}a_{21}=0$ のときは逆行列は存在しない。

\[ \pmb{\textit{A}}^{-1}\pmb{\textit{A}} = \frac{1}{a_{11}a_{22}-a_{12}a_{21}} \left( \begin{array}{cc} a_{22} & -a_{12}\\ -a_{21} & a_{11} \end{array} \right) \left( \begin{array}{cc} a_{11} & a_{12}\\ a_{21} & a_{22} \end{array} \right) =\left( \begin{array}{cc} 1 & 0\\ 0 & 1 \end{array} \right) =\pmb{\textit{I}} \]

\[ \pmb{\textit{A}}\pmb{\textit{A}}^{-1} = \pmb{\textit{I}} \]