行列の和・差の具体例

■問題1

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

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

■解答

(1)

${\displaystyle 5A}$${\displaystyle =5\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 6\hfill & \hfill 3\hfill \end{array}\right)}$${\displaystyle =\left(\begin{array}{ccc}\hfill 5\hfill & \hfill 0\hfill & \hfill -20\hfill \\ \hfill 10\hfill & \hfill 30\hfill & \hfill 15\hfill \end{array}\right)}$

(2)

${\displaystyle A+3B}$${\displaystyle =\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 6\hfill & \hfill 3\hfill \end{array}\right)+3\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill -5\hfill \end{array}\right)}$

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

${\displaystyle =\left(\begin{array}{ccc}\hfill 10\hfill & \hfill 6\hfill & \hfill 20\hfill \\ \hfill 2\hfill & \hfill 18\hfill & \hfill -12\hfill \end{array}\right)}$

(3)

${\displaystyle 2\left(3A-B\right)-3\left(A+5B\right)}$${\displaystyle =6A-2B-3A-15B}$

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

${\displaystyle =3\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 6\hfill & \hfill 3\hfill \end{array}\right)-17\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 8\hfill \\ \hfill 0\hfill & \hfill 4\hfill & \hfill -5\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 0\hfill & \hfill -12\hfill \\ \hfill 6\hfill & \hfill 18\hfill & \hfill 9\hfill \end{array}\right)-\left(\begin{array}{ccc}\hfill 51\hfill & \hfill 34\hfill & \hfill 136\hfill \\ \hfill 0\hfill & \hfill 68\hfill & \hfill -85\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{ccc}\hfill -48\hfill & \hfill -34\hfill & \hfill -148\hfill \\ \hfill 6\hfill & \hfill -50\hfill & \hfill 94\hfill \end{array}\right)}$

■問題2

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

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

■解答

(1)

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

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

${\displaystyle =\left(\begin{array}{cc}\hfill -15\hfill & \hfill -10\hfill \\ \hfill 25\hfill & \hfill 0\hfill \end{array}\right)+\left(\begin{array}{cc}\hfill 12\hfill & \hfill 21\hfill \\ \hfill 3\hfill & \hfill 9\hfill \end{array}\right)}$

${\displaystyle =\left(\begin{array}{cc}\hfill -3\hfill & \hfill 11\hfill \\ \hfill 28\hfill & \hfill 9\hfill \end{array}\right)}$

(2)

${\displaystyle 3\left(3A-B-X\right)}$${\displaystyle =4\left(A+9B+5X\right)}$

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

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

${\displaystyle 23X}$${\displaystyle =5\left(\begin{array}{cc}\hfill 3\hfill & \hfill 2\hfill \\ \hfill -5\hfill & \hfill 0\hfill \end{array}\right)-39\left(\begin{array}{cc}\hfill 4\hfill & \hfill 7\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right)}$

${\displaystyle 23X}$${\displaystyle =\left(\begin{array}{cc}\hfill 15\hfill & \hfill 10\hfill \\ \hfill -25\hfill & \hfill 0\hfill \end{array}\right)-\left(\begin{array}{cc}\hfill 156\hfill & \hfill 273\hfill \\ \hfill 39\hfill & \hfill 117\hfill \end{array}\right)}$

${\displaystyle 23X}$${\displaystyle =\left(\begin{array}{cc}\hfill -141\hfill & \hfill -263\hfill \\ \hfill -64\hfill & \hfill -117\hfill \end{array}\right)}$

${\displaystyle X}$${\displaystyle =\left(\begin{array}{cc}-\frac{141}{23}& -\frac{263}{23}\\ -\frac{64}{23}& -\frac{117}{23}\end{array}\right)}$

■問題3

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

${\displaystyle \{\begin{array}{c}\hfill X+2Y=5A\hfill \\ \hfill 2X-3Y=7B\hfill \end{array}}$

■解答

${\displaystyle X+2Y=5A}$ 　･･････(1)

${\displaystyle 2X-3Y=7B}$ 　･･････(2)

(1)×3+(2)×2より

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

(1)×2−(2)より

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

(3)より

${\displaystyle =\left(\begin{array}{ccc}\hfill 119\hfill & \hfill 17\hfill & \hfill 113\hfill \\ \hfill -9\hfill & \hfill 70\hfill & \hfill 67\hfill \end{array}\right)}$

${\displaystyle X}$${\displaystyle =\left(\begin{array}{ccc}17& \frac{17}{7}& \frac{113}{7}\\ -\frac{9}{7}& 10& \frac{67}{7}\end{array}\right)}$

(4)より

${\displaystyle =\left(\begin{array}{ccc}\hfill 63\hfill & \hfill 44\hfill & \hfill -39\hfill \\ \hfill 92\hfill & \hfill -35\hfill & \hfill -86\hfill \end{array}\right)}$

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

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最終更新日： 2025年1月17日