数学ガール　行列が描くもの　第3章の問題3-2　任意の2×2行列　A,B,C　に対して （A+B)C＝AC+BC が成り立つことの証明。

$$A= \begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{pmatrix}$$

$$B= \begin{pmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ \end{pmatrix}$$

$$C= \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix}$$

とする。

(A+B)C

$$\begin{pmatrix}\begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{pmatrix} + \begin{pmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ \end{pmatrix} \end{pmatrix} \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix}$$

$$= \begin{pmatrix} a_{11} + b_{11} & a_{12} +b_{12}\\ a_{21} + b_{21} & a_{22} + b_{22}\\ \end{pmatrix} \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix}$$

$$= \begin{pmatrix}\begin{pmatrix} a_{11} + b_{11} \end{pmatrix}c_{11} + \begin{pmatrix}a_{12} + b_{12} \end{pmatrix}c_{21} & \begin{pmatrix}a_{11} + b_{11} \end{pmatrix}c_{12} + \begin{pmatrix}a_{12} + b_{12} \end{pmatrix}c_{22} \\ \begin{pmatrix}a_{21} + b_{21} \end{pmatrix}c_{11} + \begin{pmatrix}a_{22} + b_{22} \end{pmatrix}c_{21} & \begin{pmatrix}a_{21} + b_{21} \end{pmatrix}c_{12} + \begin{pmatrix}a_{22} + b_{22} \end{pmatrix}c_{22} \\ \end{pmatrix}$$

$$= \begin{pmatrix} a_{11}c_{11} + b_{11}c_{11} + a_{12}c_{21} + b_{12}c_{21} & a_{11}c_{12} + b_{11}c_{12} + a_{12}c_{22} + b_{12}c_{22} \\ a_{21}c_{11} + b_{21}c_{11} + a_{22}c_{21} + b_{22}c_{21} & a_{21}c_{12} + b_{21}c_{12} + a_{22}c_{22} + b_{22}c_{22} \\ \end{pmatrix}$$

右辺AC+BC

$$\begin{pmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{pmatrix} \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix} + \begin{pmatrix} b_{11} & b_{12}\\ b_{21} & b_{22}\\ \end{pmatrix} \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix}$$

$$= \begin{pmatrix} a_{11}c_{11} + a_{12}c_{21} & a_{11}c_{12} +a_{12}c_{22}\\ a_{21}c_{11} + a_{22}c_{21} & a_{21}c_{12} +a_{22}c_{22} \end{pmatrix} + \begin{pmatrix} b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} +b_{12}c_{22}\\ b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} +b_{22}c_{22} \end{pmatrix}$$

$$= \begin{pmatrix} a_{11}c_{11} + a_{12}c_{21} + b_{11}c_{11} + b_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} + b_{11}c_{12} + b_{12}c_{22}\\ a_{21}c_{11} + a_{22}c_{21} + b_{21}c_{11} + b_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} + b_{21}c_{12} + b_{22}c_{22} \end{pmatrix}$$

左辺と右辺を比較

左辺

$$(A+B)C= \begin{pmatrix} a_{11}c_{11} + b_{11}c_{11} + a_{12}c_{21} + b_{12}c_{21} & a_{11}c_{12} + b_{11}c_{12} + a_{12}c_{22} + b_{12}c_{22} \\ a_{21}c_{11} + b_{21}c_{11} + a_{22}c_{21} + b_{22}c_{21} & a_{21}c_{12} + b_{21}c_{12} + a_{22}c_{22} + b_{22}c_{22} \\ \end{pmatrix}$$

右辺

$$AC+BC= \begin{pmatrix} a_{11}c_{11} + a_{12}c_{21} + b_{11}c_{11} + b_{12}c_{21} & a_{11}c_{12} + a_{12}c_{22} + b_{11}c_{12} + b_{12}c_{22}\\ a_{21}c_{11} + a_{22}c_{21} + b_{21}c_{11} + b_{22}c_{21} & a_{21}c_{12} + a_{22}c_{22} + b_{21}c_{12} + b_{22}c_{22} \end{pmatrix}$$

計算した結果、成分が等しいので(A+B)C=AC+BCが成り立つ。