# 行列の計算

1. 次の行列の計算をせよ．

(1) $\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)+\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)$   解答

(2) $\left(\begin{array}{c}\begin{array}{ccc}3& 7& 8\end{array}\\ \begin{array}{ccc}1& 5& 2\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{ccc}6& 2& 4\end{array}\\ \begin{array}{ccc}7& 1& 9\end{array}\end{array}\right)$   解答

(3) $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$   解答

(4) $\left(\begin{array}{cc}1& 3\\ 5& 7\end{array}\right)\left(\begin{array}{cc}8& 6\\ 4& 2\end{array}\right)$   解答

(5) $\left(\begin{array}{cc}2& 3\\ 4& 5\end{array}\right)\left(\begin{array}{c}\begin{array}{ccc}6& 5& 4\end{array}\\ \begin{array}{ccc}3& 2& 1\end{array}\end{array}\right)$   解答

(6) $\left(\begin{array}{ccc}a& b& c\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$    解答

(7) $\left(\begin{array}{c}a\\ b\\ c\end{array}\right)\left(\begin{array}{ccc}x& y& z\end{array}\right)$    解答

(8) $\left(\begin{array}{cc}2& 6\\ 4& 3\end{array}\right)-\left(\begin{array}{cc}3& 4\\ 1& 5\end{array}\right)$   解答

(9) $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)$$-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) $3\left(\begin{array}{c}\begin{array}{ccc}-1& 2& 1\\ 1& 3& -1\end{array}\end{array}\right)+2\left(\begin{array}{c}\begin{array}{ccc}-1& 1& 2\\ -2& 1& -5\end{array}\end{array}\right)$   解答

(11) $\left(\begin{array}{ccc}3& 1& -2\end{array}\right)\left(\begin{array}{c}2\\ -3\\ 4\end{array}\right)$   解答

(12)   $\left(\begin{array}{ccc}2& 3& -2\\ 0& -2& 5\\ 1& 1& 3\end{array}\right)\left(\begin{array}{cc}\begin{array}{c}2\\ -1\\ 2\end{array}& \begin{array}{c}1\\ 5\\ -2\end{array}\end{array}\right)$ 解答

(13) ${\left(\begin{array}{ccc}1& 0& -1\\ 1& 0& -2\\ 3& -2& 1\end{array}\right)}^{2}$   解答

2. 掃き出し法を用いて，次の連立方程式を解け．
•  (1)  $\left\{\begin{array}{c}x+3y+2z=0\\ x-4y+2z=7\\ 2x-y+5z=6\end{array}$  ⇒ 解答
•  (2) $\left\{\begin{array}{c}x-2y+4z=8\\ 3x-y+2z=9\\ 2x+y-z=6\end{array}\right\$  ⇒ 解答
•  (3)  $\left\{\begin{array}{c}x-3y+z=-2\\ -3x+4y+2z=-19\\ 2x+y-z=11\end{array}$  ⇒ 解答
•  (4)  $\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
•  (5)  $\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
•  (6)  $\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\$  ⇒
•  (7) $\left\{\begin{array}{c}x+3y+5z=6\\ 3x+7y+11z=16\\ -x+4y+7z=-3\end{array}$  ⇒
•  (8) $\left\{\begin{array}{c}3x-y-2z=2\\ 2y-z=-1\\ 3x-5y=3\end{array}$  ⇒
3. 次の等式を満たす行列を求めよ．

(1) $A=\left(\begin{array}{cc}-2& 1\\ 1& 2\end{array}\right)$$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$A-2X=\frac{1}{3}\left\{4A-\left(B+3X\right)\right\}$ を満たす行列$X$

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

(2) $A=\left(\begin{array}{cc}5& 3\\ -7& -10\end{array}\right)$$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 $\frac{2}{5}A-\frac{2}{3}X$$=\frac{1}{6}\left\{-2A-\left(2X-3B\right)\right\}$ を満たす行列$X$

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

(3) $A=\left(\begin{array}{cc}-3& 5\\ 2& 1\end{array}\right)$$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${A}^{2}+\frac{1}{2}X=\frac{1}{6}\left\{2{B}^{2}-3\left(X-2{A}^{2}\right)\right\}$ を満たす行列$X$

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

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

4. 連立方程式

$\left\{\begin{array}{c}2x+y=7\\ 5x+2y=16\end{array}$
係数行列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

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

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

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