基本的な行列の問題

■問題

$3$\displaystyle{ A=\left({\begin{array} - 2 &1& \vspace{6}\\ 1&2& \vspace{6}\\ \end{array}}\right) }$ ， $3$\displaystyle{ B=\left({\begin{array}1&2& \vspace{6}\\ - 6 &9& \vspace{6}\\ \end{array}}\right) }$ のとき，次の行列を求めよ．

[i]等式 $3$\displaystyle{ A- 2X=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left{{ 4A- 3\left({ B+3X }\right) }\right} }$ を満たす行列 $3$\displaystyle{X}$

[ii]2つの等式 $3$\displaystyle{ P+Q=A }$ ， $3$\displaystyle{ P- Q=B }$ を同時に満たす2次の正方行列 $3$\displaystyle{P}$ ， $3$\displaystyle{Q}$

■解答

[i]等式 $3$\displaystyle{ A- 2X=\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left{{ 4A- 3\left({ B+3X }\right) }\right} }$ を満たす行列 $3$\displaystyle{X}$

$3$\displaystyle{ A- 2X }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left{{ 4A- 3\left({ B+3X }\right) }\right} }$

$3$\displaystyle{ 3A- 6X }$ $3$\displaystyle{ =4A- 3\left({ B+3X }\right) }$

$3$\displaystyle{ 3A- 6X }$ $3$\displaystyle{ =4A- 3B- 9X }$

$3$\displaystyle{ 3X }$ $3$\displaystyle{ =A- 3B }$

$3$\displaystyle{ X }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left({ A- 3B }\right) }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left{{ \left({\begin{array} - 2 &1& \vspace{6}\\ 1&2& \vspace{6}\\ \end{array}}\right)- 3\left({\begin{array}1&2& \vspace{6}\\ - 6 &9& \vspace{6}\\ \end{array}}\right) }\right} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left{{ \left({\begin{array} - 2 &1& \vspace{6}\\ 1&2& \vspace{6}\\ \end{array}}\right)+\left({\begin{array} - 3 & - 6 & \vspace{6}\\ 18& - 27 & \vspace{6}\\ \end{array}}\right) }\right} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}}\left({\begin{array} - 5 & - 5 & \vspace{6}\\ 19& - 25 & \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({\begin{array} - \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}} & - \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}} & \vspace{6}\\ \frac{\hspace{2}19\hspace{2}}{\hspace{2}3\hspace{2}}& - \frac{\hspace{2}25\hspace{2}}{\hspace{2}3\hspace{2}} & \vspace{6}\\ \end{array}}\right) }$

[ii]2つの等式 $3$\displaystyle{ P+Q=A }$ ， $3$\displaystyle{ P- Q=B }$ を同時に満たす2次の正方行列 $3$\displaystyle{P}$ ， $3$\displaystyle{Q}$

$3$\displaystyle{ P+Q=A }$ 　･･････(1)

$3$\displaystyle{ P-Q=B }$ 　･･････(2)

(1)+(2)，(1)-(2)の計算をすると

$3$\displaystyle{ 2P=A+B }$ ， $3$\displaystyle{ 2Q=A- B }$

$3$\displaystyle{ P }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left({ A+B }\right) }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \left({\begin{array} - 2 &1& \vspace{6}\\ 1&2& \vspace{6}\\ \end{array}}\right)+\left({\begin{array}1&2& \vspace{6}\\ - 6 &9& \vspace{6}\\ \end{array}}\right) }\right} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left({\begin{array} - 1 &3& \vspace{6}\\ - 5 &11& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({\begin{array} - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} &\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}& \vspace{6}\\ - \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}} &\frac{\hspace{2}11\hspace{2}}{\hspace{2}2\hspace{2}}& \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ Q }$ $3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left({ A- B }\right) }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left{{ \left({\begin{array} - 2 &1& \vspace{6}\\ 1&2& \vspace{6}\\ \end{array}}\right)- \left({\begin{array}1&2& \vspace{6}\\ - 6 &9& \vspace{6}\\ \end{array}}\right) }\right} }$

$3$\displaystyle{ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\left({\begin{array} - 3 & - 1 & \vspace{6}\\ 7& - 7 & \vspace{6}\\ \end{array}}\right) }$

$3$\displaystyle{ =\left({\begin{array} - \frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}} & - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} & \vspace{6}\\ \frac{\hspace{2}7\hspace{2}}{\hspace{2}2\hspace{2}}& - \frac{\hspace{2}7\hspace{2}}{\hspace{2}2\hspace{2}} & \vspace{6}\\ \end{array}}\right) }$

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最終更新日： 2022年6月17日