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

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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。

左辺(AB)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_{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{pmatrix} \begin{pmatrix} c_{11} & c_{12}\\ c_{21} & c_{22}\\ \end{pmatrix}
= \begin{pmatrix}\begin{pmatrix} a_{11}b_{11} + a_{12}b_{21}\end{pmatrix} c_{11} + \begin{pmatrix} a_{11}b_{12} + a_{12}b_{22}\end{pmatrix} c_{21} & \begin{pmatrix}a_{11}b_{11} + a_{12}b_{21}\end{pmatrix} c_{12} + \begin{pmatrix}a_{11}b_{12} + a_{12}b_{22}\end{pmatrix} c_{22} \\ \begin{pmatrix} a_{21}b_{11} + a_{22}b_{21}\end{pmatrix} c_{11} + \begin{pmatrix} a_{21}b_{12} + a_{22}b_{22}\end{pmatrix} c_{21} & \begin{pmatrix}a_{21}b_{11} + a_{22}b_{21}\end{pmatrix} c_{12} + \begin{pmatrix}a_{21}b_{12} + a_{22}b_{22}\end{pmatrix} c_{22} \end{pmatrix}
= \begin{pmatrix} a_{11}b_{11}c_{11} + a_{12}b_{21}c_{11} + a_{11}b_{12} c_{21} + a_{12}b_{22} c_{21} & a_{11}b_{11}c_{12} + a_{12}b_{21}c_{12} + a_{11}b_{12} c_{22} + a_{12}b_{22} c_{22} \\ a_{21}b_{11}c_{11} + a_{22}b_{21}c_{11} + a_{21}b_{12}c_{21} + a_{22}b_{22}c_{21} &a_{21}b_{11}c_{12} + a_{22}b_{21}c_{12} + a_{21}b_{12}c_{22} + a_{22}b_{22}c_{22} \end{pmatrix}

右辺A(BC)

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

左辺と右辺を比較

左辺

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

右辺

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

計算の結果、成分が同一なので(AB)C=A(BC)が成り立つ。

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