行列の積の具体例

${\displaystyle A=\left(\begin{array}{lll}1\hfill & 5\hfill & 9\hfill \end{array}\right)}$ , ${\displaystyle B=\left(\begin{array}{l}5\hfill \\ 2\hfill \end{array}\right)}$ , ${\displaystyle C=\left(\begin{array}{lll}7\hfill & 4\hfill & -1\hfill \\ 3\hfill & 0\hfill & 5\hfill \end{array}\right)}$ , ${\displaystyle D=\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)}$ , ${\displaystyle E=\left(\begin{array}{lll}8\hfill & 4\hfill & 2\hfill \\ 1\hfill & 3\hfill & -6\hfill \\ -7\hfill & 0\hfill & 5\hfill \end{array}\right)}$ のうち，2つの行列が定義されるものについて，その積を求めてみる．

計算:

${\displaystyle AD}$${\displaystyle =\left(\begin{array}{lll}1\hfill & 5\hfill & 9\hfill \end{array}\right)\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ll}56\hfill & -2\hfill \end{array}\right)}$

${\displaystyle AE}$${\displaystyle =\left(\begin{array}{lll}1\hfill & 5\hfill & 9\hfill \end{array}\right)\left(\begin{array}{lll}8\hfill & 4\hfill & 2\hfill \\ 1\hfill & 3\hfill & -6\hfill \\ -7\hfill & 0\hfill & 5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{lll}-50\hfill & 19\hfill & 17\hfill \end{array}\right)}$

${\displaystyle BA}$${\displaystyle =\left(\begin{array}{l}5\hfill \\ 2\hfill \end{array}\right)\left(\begin{array}{lll}1\hfill & 5\hfill & 9\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{lll}5\hfill & 25\hfill & 45\hfill \\ 2\hfill & 10\hfill & 18\hfill \end{array}\right)}$

${\displaystyle CD}$${\displaystyle =\left(\begin{array}{lll}7\hfill & 4\hfill & -1\hfill \\ 3\hfill & 0\hfill & 5\hfill \end{array}\right)\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ll}35+12-4\hfill & 14+4+1\hfill \\ 15+0+20\hfill & 6+0-5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ll}43\hfill & 19\hfill \\ 35\hfill & 1\hfill \end{array}\right)}$

${\displaystyle CE}$${\displaystyle =\left(\begin{array}{lll}7\hfill & 4\hfill & -1\hfill \\ 3\hfill & 0\hfill & 5\hfill \end{array}\right)\left(\begin{array}{lll}8\hfill & 4\hfill & 2\hfill \\ 1\hfill & 3\hfill & -6\hfill \\ -7\hfill & 0\hfill & 5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{lll}67\hfill & 40\hfill & -15\hfill \\ -11\hfill & 12\hfill & 31\hfill \end{array}\right)}$

${\displaystyle DB}$${\displaystyle =\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)\left(\begin{array}{l}5\hfill \\ 2\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{l}29\hfill \\ 17\hfill \\ 18\hfill \end{array}\right)}$

${\displaystyle DC}$${\displaystyle =\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)\left(\begin{array}{lll}7\hfill & 4\hfill & -1\hfill \\ 3\hfill & 0\hfill & 5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{lll}35+6\hfill & 20+0\hfill & -5+10\hfill \\ 21+3\hfill & 12+0\hfill & -3+5\hfill \\ 28-3\hfill & 16+0\hfill & -4-5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{lll}41\hfill & 20\hfill & 5\hfill \\ 24\hfill & 12\hfill & 2\hfill \\ 25\hfill & 16\hfill & -9\hfill \end{array}\right)}$

${\displaystyle ED}$${\displaystyle =\left(\begin{array}{lll}8\hfill & 4\hfill & 2\hfill \\ 1\hfill & 3\hfill & -6\hfill \\ -7\hfill & 0\hfill & 5\hfill \end{array}\right)\left(\begin{array}{ll}5\hfill & 2\hfill \\ 3\hfill & 1\hfill \\ 4\hfill & -1\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ll}40+12+8\hfill & 16+4-2\hfill \\ 5+9-24\hfill & 2+3+6\hfill \\ -35+0+20\hfill & -14+0-5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ll}60\hfill & 18\hfill \\ 21\hfill & 11\hfill \\ -15\hfill & -19\hfill \end{array}\right)}$

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最終更新日： 2023年2月9日