行列の計算

1. 次の行列の計算をせよ．
   1. ${\displaystyle \left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)+\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)}$   ⇒ 解答
   2. ${\displaystyle \left(\begin{array}{c}\begin{array}{ccc}3& 7& 8\end{array}\\ \begin{array}{ccc}1& 5& 2\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{ccc}6& 2& 4\end{array}\\ \begin{array}{ccc}7& 1& 9\end{array}\end{array}\right)}$   ⇒ 解答
   3. ${\displaystyle \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)}$   ⇒ 解答
   4. ${\displaystyle \left(\begin{array}{cc}1& 3\\ 5& 7\end{array}\right)\left(\begin{array}{cc}8& 6\\ 4& 2\end{array}\right)}$   ⇒ 解答
   5. ${\displaystyle \left(\begin{array}{cc}2& 3\\ 4& 5\end{array}\right)\left(\begin{array}{c}\begin{array}{ccc}6& 5& 4\end{array}\\ \begin{array}{ccc}3& 2& 1\end{array}\end{array}\right)}$   ⇒ 解答
   6. ${\displaystyle \left(\begin{array}{ccc}a& b& c\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)}$   ⇒ 解答
   7. ${\displaystyle \left(\begin{array}{c}a\\ b\\ c\end{array}\right)\left(\begin{array}{ccc}x& y& z\end{array}\right)}$   ⇒ 解答
   8. ${\displaystyle \left(\begin{array}{cc}2& 6\\ 4& 3\end{array}\right)-\left(\begin{array}{cc}3& 4\\ 1& 5\end{array}\right)}$   ⇒ 解答
   9. ${\displaystyle 3\left(\begin{array}{ccc}\begin{array}{c}4\\ 0\end{array}& \begin{array}{c}2\\ 5\end{array}& \begin{array}{c}-3\\ -1\end{array}\end{array}\right)}$ ${\displaystyle -2\left(\begin{array}{ccc}\begin{array}{c}3\\ 4\end{array}& \begin{array}{c}1\\ -1\end{array}& \begin{array}{c}-2\\ 0\end{array}\end{array}\right)}$   ⇒ 解答
   10. ${\displaystyle 3\left(\begin{array}{c}\begin{array}{ccc}-1& 2& 1\\ 1& 3& -1\end{array}\end{array}\right)+2\left(\begin{array}{c}\begin{array}{ccc}-1& 1& 2\\ -2& 1& -5\end{array}\end{array}\right)}$   ⇒ 解答
   11. ${\displaystyle \left(\begin{array}{ccc}3& 1& -2\end{array}\right)\left(\begin{array}{c}2\\ -3\\ 4\end{array}\right)}$   ⇒ 解答
   12. ${\displaystyle \left(\begin{array}{ccc}2& 3& -2\\ 0& -2& 5\\ 1& 1& 3\end{array}\right)\left(\begin{array}{cc}\begin{array}{c}2\\ -1\\ 2\end{array}& \begin{array}{c}1\\ 5\\ -2\end{array}\end{array}\right)}$ ⇒ 解答
   13. ${\displaystyle {\left(\begin{array}{ccc}1& 0& -1\\ 1& 0& -2\\ 3& -2& 1\end{array}\right)}^2}$   ⇒ 解答
2. 掃き出し法を用いて，次の連立方程式を解け．

   • (*)  ${\displaystyle \{\begin{array}{c}x+3y+2z=0\\ x-4y+2z=7\\ 2x-y+5z=6\end{array}}$  ⇒ 解答

   • (*)  ${\displaystyle \left\{\begin{array}{c}x-2y+4z=8\\ 3x-y+2z=9\\ 2x+y-z=6\end{array}\right.}$  ⇒ 解答

   • (*)  ${\displaystyle \{\begin{array}{c}x-3y+z=-2\\ -3x+4y+2z=-19\\ 2x+y-z=11\end{array}}$  ⇒ 解答

   • (*)  ${\displaystyle \left\{\begin{array}{c}a-b-2c+\mathrm{2d}=4\\ 2a+b-3c-d=-3\\ -a+2b+2c-2c=-6\\ 3a-3b+c+d=7\end{array}\right.}$  ⇒ 解答

   • (*)  ${\displaystyle \left\{\begin{array}{c}a+b-2c-2d=-5\\ 2a-2b+c-d=8\\ -a+b-c+2d=-5\\ -2a-b+c-3d=2\end{array}\right.}$  ⇒ 解答

   • (*)  ${\displaystyle {\displaystyle \left\{\begin{array}{c}a+2b+d-e=-4\\ -a-b-c+d=3\\ -2b+2c+d-e=-1\\ a+b+c-2d+e=-2\\ a-b-2c+2e=9\end{array}\right.}}$  ⇒ 解答

   • (*)  ${\displaystyle \{\begin{array}{c}x+3y+5z=6\\ 3x+7y+11z=16\\ -x+4y+7z=-3\end{array}}$  ⇒ 解答

   • (*)  ${\displaystyle \{\begin{array}{c}3x-y-2z=2\\ 2y-z=-1\\ 3x-5y=3\end{array}}$  ⇒ 解答
3. 次の等式を満たす行列を求めよ．
   1. ${\displaystyle A=\left(\begin{array}{cc}-2& 1\\ 1& 2\end{array}\right)}$ ， ${\displaystyle B=\left(\begin{array}{cc}1& 2\\ -6& 9\end{array}\right)}$ のとき，次の行列を求めよ．

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

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

   2. ${\displaystyle A=\left(\begin{array}{cc}5& 3\\ -7& -10\end{array}\right)}$ ， ${\displaystyle B=\left(\begin{array}{cc}-2& 5\\ 1& -9\end{array}\right)}$ のとき，次の行列を求めよ．

      [i] 等式 ${\displaystyle \frac{2}{5}A-\frac{2}{3}X}$ ${\displaystyle =\frac{1}{6}\left\{-2A-\left(2X-3B\right)\right\}}$ を満たす行列 ${\displaystyle X}$

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

   3. ${\displaystyle A=\left(\begin{array}{cc}-3& 5\\ 2& 1\end{array}\right)}$ ， ${\displaystyle B=\left(\begin{array}{cc}2& -3\\ -1& 7\end{array}\right)}$ のとき，次の行列を求めよ．

      [i]等式 ${\displaystyle \frac{1}{3}{A}^2-X=\frac{1}{6}\left\{2B^2-4\left(X-2A^2\right)\right\}}$ を満たす行列 ${\displaystyle X}$

      [ii]2つの等式 ${\displaystyle 2\left(P+Q\right)=A}$ ， ${\displaystyle \frac{1}{3}\left(P-Q\right)=B}$ を同時に満たす2次の正方行列 ${\displaystyle P}$ ， ${\displaystyle Q}$  ⇒   解答

   4. ${\displaystyle A=\frac{1}{3}\left(\begin{array}{cc}9& 1\\ -6& -1\end{array}\right)}$ ， ${\displaystyle B=\left(\begin{array}{cc}3& 1\\ -5& -3\end{array}\right)}$ のとき， ${\displaystyle A{B}^5{A}^{-1}}$ を求めよ． ⇒   解答
4. 連立方程式

   　　 ${\displaystyle \{\begin{array}{c}2x+y=7\\ 5x+2y=16\end{array}}$
   を係数行列の逆行列を利用して解け．計算過程を書くこと． ⇒解答

5. 掃き出し法を用いて

   　　 ${\displaystyle A=\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 2\\ 1& 3& 4\end{array}\right)}$
   の逆行列 ${\displaystyle A^{-1}}$ を求めよ． ⇒解答

6. 平面 ${\displaystyle 2x-3y+z=1}$ と平面 ${\displaystyle 3x+2y-3z=1}$ の交線の方程式を求めよ． ⇒解答

 

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最終更新日：2025年11月9日