行列の和・差の具体例

■問題1

$3$\displaystyle{A= $ \begin{array}1&0& - 4 & \vspace{6}\\ 2&6&3& \vspace{6}\\ \end{array} $ }$ , $3$\displaystyle{B= $ \begin{array}3&2&8& \vspace{6}\\ 0&4& - 5 & \vspace{6}\\ \end{array} $ }$ のとき，次の行列を求めよ．

(1) $3$\displaystyle{5A}$ (2) $3$\displaystyle{A+3B}$ (3) $3$\displaystyle{2 $ 3A- B $ - 3$A+5B$}$

■解答

(1)

$3$\displaystyle{ 5A }$ $3$\displaystyle{ =5 $ \begin{array}1&0& - 4 & \vspace{6}\\ 2&6&3& \vspace{6}\\ \end{array} $ }$ $3$\displaystyle{= $ \begin{array}5&0& - 20 & \vspace{6}\\ 10 & 30 & 15 & \vspace{6}\\ \end{array} $ }$

(2)

$3$\displaystyle{A+3B}$ $3$\displaystyle{}$ $3$\displaystyle{= $ \begin{array}1&0& - 4 & \vspace{6}\\ 2&6&3& \vspace{6}\\ \end{array} $ +3 $ \begin{array}3&2&8& \vspace{6}\\ 0&4& - 5 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} 1+3\cdot 3 & 0+3\cdot 2 & - 4+3\cdot 8 & \vspace{6}\\ 2+3\cdot 0 & 6+3\cdot 4 & 3+3\cdot \( - 5 $ & \vspace{6}\\ \end{array} \) }$

$3$\displaystyle{= $ \begin{array} 10 &6& 20 & \vspace{6}\\ 2& 18 & - 12 & \vspace{6}\\ \end{array} $ }$

(3)

$3$\displaystyle{2 $ 3A- B $ - 3 $ A+5B $ }$ $3$\displaystyle{=6A- 2B- 3A- 15B}$

$3$\displaystyle{=3A- 17B}$

$3$\displaystyle{=3 $ \begin{array}1&0& - 4 & \vspace{6}\\ 2&6&3& \vspace{6}\\ \end{array} $ - 17 $ \begin{array}3&2&8& \vspace{6}\\ 0&4& - 5 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array}3&0& - 12 & \vspace{6}\\ 6& 18 &9& \vspace{6}\\ \end{array} $ - $ \begin{array} 51 & 34 & 136 & \vspace{6}\\ 0& 68 & - 85 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} - 48 & - 34 & - 148 & \vspace{6}\\ 6& - 50 & 94 & \vspace{6}\\ \end{array} $ }$

■問題2

$3$\displaystyle{A= $ \begin{array}3&2& \vspace{6}\\ - 5 &0& \vspace{6}\\ \end{array} $ }$ , $3$\displaystyle{B= $ \begin{array}4&7& \vspace{6}\\ 1&3& \vspace{6}\\ \end{array} $ }$ のとき，次の各等式を満たす行列 $3$\displaystyle{X}$ を求めよ．

(1) $3$\displaystyle{5A+X=3B}$ (2) $3$\displaystyle{\frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}} $ 3A- B- X $ =\frac{\hspace{2}1\hspace{2}}{\hspace{2}3\hspace{2}} $ A+9B+5X $ }$

■解答

(1)

$3$\displaystyle{X}$ $3$\displaystyle{=- 5A+3B}$

$3$\displaystyle{=- 5 $ \begin{array}3&2& \vspace{6}\\ - 5 &0& \vspace{6}\\ \end{array} $ +3 $ \begin{array}4&7& \vspace{6}\\ 1&3& \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} - 15 & - 10 & \vspace{6}\\ 25 &0& \vspace{6}\\ \end{array} $ + $ \begin{array} 12 & 21 & \vspace{6}\\ 3&9& \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} - 3 & 11 & \vspace{6}\\ 28 &9& \vspace{6}\\ \end{array} $ }$

(2)

$3$\displaystyle{3 $ 3A- B- X $ }$ $3$\displaystyle{=4 $ A+9B+5X $ }$

$3$\displaystyle{9A- 3B- 3X}$ $3$\displaystyle{=4A+36B+20X}$

$3$\displaystyle{23X}$ $3$\displaystyle{=5A- 39B}$

$3$\displaystyle{23X}$ $3$\displaystyle{=5 $ \begin{array}3&2& \vspace{6}\\ - 5 &0& \vspace{6}\\ \end{array} $ - 39 $ \begin{array}4&7& \vspace{6}\\ 1&3& \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{23X}$ $3$\displaystyle{= $ \begin{array} 15 & 10 & \vspace{6}\\ - 25 &0& \vspace{6}\\ \end{array} $ - $ \begin{array} 156 & 273 & \vspace{6}\\ 39 & 117 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{23X}$ $3$\displaystyle{= $ \begin{array} - 141 & - 263 & \vspace{6}\\ - 64 & - 117 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{X}$ $3$\displaystyle{ =\left({ \begin{array} - \frac{\hspace{2} 141 \hspace{2}}{\hspace{2} 23 \hspace{2}} & - \frac{\hspace{2} 263 \hspace{2}}{\hspace{2} 23 \hspace{2}} & \vspace{6}\\ - \frac{\hspace{2} 64 \hspace{2}}{\hspace{2} 23 \hspace{2}} & - \frac{\hspace{2} 117 \hspace{2}}{\hspace{2} 23 \hspace{2}} & \vspace{6}\\ \end{array} }\right) }$

■問題3

$3$\displaystyle{A= $ \begin{array}7&3&1& \vspace{6}\\ 5&0& - 3 & \vspace{6}\\ \end{array} $ }$ , $3$\displaystyle{B= $ \begin{array}1& - 2 &7& \vspace{6}\\ - 6 &5&8& \vspace{6}\\ \end{array} $ }$ のとき，次の等式を満たす行列 $3$\displaystyle{X}$ , $3$\displaystyle{Y}$ を求めよ．

$3$\displaystyle{ \{ \begin{array} X+2Y=5A & \vspace{6}\\ 2X- 3Y=7B & \vspace{6}\\ \end{array} }$

■解答

$3$\displaystyle{ X+2Y=5A }$ 　･･････(1)

$3$\displaystyle{ 2X- 3Y=7B }$ 　･･････(2)

(1)×3+(2)×2より

$3$\displaystyle{7X}$ $3$\displaystyle{=15A+14B}$ 　･･････(3)

(1)×2−(2)より

$3$\displaystyle{7Y}$ $3$\displaystyle{=10A- 7B}$ 　･･････(4)

(3)より

$3$\displaystyle{7X}$ $3$\displaystyle{=15 $ \begin{array}7&3&1& \vspace{6}\\ 5&0& - 3 & \vspace{6}\\ \end{array} $ +14 $ \begin{array}1& - 2 &7& \vspace{6}\\ - 6 &5&8& \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} 105 & 45 & 15 & \vspace{6}\\ 75 &0& - 45 & \vspace{6}\\ \end{array} $ + $ \begin{array} 14 & - 28 & 98 & \vspace{6}\\ - 84 & 70 & 112 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} 119 & 17 & 113 & \vspace{6}\\ - 9 & 70 & 67 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{X}$ $3$\displaystyle{ =\left({ \begin{array} 17 & \frac{\hspace{2} 17 \hspace{2}}{\hspace{2}7\hspace{2}} & \frac{\hspace{2} 113 \hspace{2}}{\hspace{2}7\hspace{2}} & \vspace{6}\\ - \frac{\hspace{2}9\hspace{2}}{\hspace{2}7\hspace{2}} & 10 & \frac{\hspace{2} 67 \hspace{2}}{\hspace{2}7\hspace{2}} & \vspace{6}\\ \end{array} }\right) }$

(4)より

$3$\displaystyle{7Y}$ $3$\displaystyle{=10 $ \begin{array}7&3&1& \vspace{6}\\ 5&0& - 3 & \vspace{6}\\ \end{array} $ - 7 $ \begin{array}1& - 2 &7& \vspace{6}\\ - 6 &5&8& \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} 70 & 30 & 10 & \vspace{6}\\ 50 &0& - 30 & \vspace{6}\\ \end{array} $ - $ \begin{array}7& - 14 & 49 & \vspace{6}\\ - 42 & 35 & 56 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \begin{array} 63 & 44 & - 39 & \vspace{6}\\ 92 & - 35 & - 86 & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{Y}$ $3$\displaystyle{ =\left({ \begin{array}9& \frac{\hspace{2} 44 \hspace{2}}{\hspace{2}7\hspace{2}} & - \frac{\hspace{2} 39 \hspace{2}}{\hspace{2}7\hspace{2}} & \vspace{6}\\ \frac{\hspace{2} 92 \hspace{2}}{\hspace{2}7\hspace{2}} & - 5 & - \frac{\hspace{2} 86 \hspace{2}}{\hspace{2}7\hspace{2}} & \vspace{6}\\ \end{array} }\right) }$

ホーム>>カテゴリー分類>>行列>>行列の和，差>>行列の和，差の具体例

学生スタッフ作成

最終更新日： 2025年1月17日