![]() ![]() *** 大区分 *** 数Ⅰ・A数Ⅱ・B数Ⅲ高卒・大学初年度 *** 中区分 *** ベクトル・行列連立方程式複素数関数・数列微分積分微分方程式統計maxima ※高卒から大学初年度向け「連立方程式」について,このサイトには次の教材があります.
この頁へGoogleやYAHOO ! などの検索から直接来てしまったので「前提となっている内容が分からない」という場合や「この頁は分かったがもっと応用問題を見たい」という場合は,他の頁を見てください. が現在地です. ↓表計算オートフィルの使い方 ↓Excelの行列計算(1) 行列の積 ↓未知数がn個で逆行列を用いて解く場合 ↓クラメールの公式で解く場合-現在地 ↓クラメールの公式[問題] ↓wxMaximaで解く場合 ↓不能解・不定解が出る場合 ↓同(固有ベクトル) ↓行基本変形と掃き出し法による場合 ↓同(不定解) ↓連立方程式(まとめ) 2次2次連立形 |
(はじめに) ○数学の歴史において,行列式の考えは連立1次方程式を解く作業の中で生まれた.(ライプニッツ,クラメールなど) ○この頁では,行列式の性質を使ってクラメールの公式を証明する.また,クラメールの公式を使って連立方程式が解けるように練習する. 【未知数が2個の連立1次方程式の場合】 ![]() cx+dy=q すなわち の解は(ad−bc≠0のとき) となるが,この式の分母および分子を今日の行列式の記号を使って書くと となる.これがクラメールの公式と呼ばれるものである. 【未知数が3個の連立1次方程式の場合】 ![]() dx+ey+fz=q gx+hy+iz=r すなわち の解をクラメールの公式を使って書くと(分母が0でないとき) |
(行列式の性質) (1) 行列式は,正方行列に対してだけ定義され,行数と列数が異なる行列に対しては定義されない. (2) 行列式は正方行列に対して定義される関数で,正方行列をAとするときdet(A),D(A)もしくは|A|で表す. さらに以下においては,行列Aを構成している列ベクトルごとに取り扱うこととし,
2次の場合,
とおくと,行列式は などの記号で書ける. 3次の場合は, のとき, とおくと, 行列式は などと書ける. (3) 行列式はn次正方行列に対して1つの数を対応させる関数で次の3つの条件を満たす.
1(多重線形性)・・・どの列に対しても線形である
○線形代数の教科書には,上記の1(多重線形性),2(交代性),3(正規性)の3つの条件を満たすものは,われわれが知っている行列式だけであることが示されている.(証明はかなり長いものになるので,この教材では省略する.)
例えば,1列目が2つの列ベクトルの和であるとき
2(交代性)・・・2つの列を入れ換えると符号が変わるまた,1列目がある列ベクトルの定数倍であるとき
例えば,1列目と2列目を入れ換えたとき
3(正規性)・・・単位行列の行列式は1であるまた,2列目と3列目を入れ換えたとき (※)なお,転置行列の行列式は元の行列の行列式に等しい
2次の行列式は,
3次の行列式は,余因子展開により2次の行列式を使って定義し ○この教材では行列式を(*1)(*2)のように定める上記の1,2,3などの性質が成り立つことを示し,これを使って行列式を変形する方法を学ぶ. 以下,具体的に示すために3次の行列式の場合を例として示すが,他の場合でも同様に示せる.
1(多重線形性)←(*2) の証明
だから が成り立つ. (※)により,行についても多重線形性が言える.
2(交代性)←(*2) の証明
まず,余因子展開を行うときは,各成分に次のようにチェック模様(市松模様)になるように符号を付けることを思い出そう.これにより, を1列目に沿って展開するときは となるのに対して,
1列目と2列目を入れ換えた場合
このように隣り合う2つの列を入れ換えると符号が変わることは容易に示せる.を2列目に沿って展開すると となる. 次に,2つ以上離れた2つの列を入れ換えるには,上記の隣り合う列の入れ換えを繰り返し行えばよいが,列番号が1つ,2つ,3つ,...と異なるに応じて,隣り合う列の項間は1回,3回,5回,...などと「奇数回」行うことになるから,結局,符号が変わる.
実際には,最小回数でなく無駄の多い入れ換え方法を経由してでも結果を合わせることができるが,その場合でも隣り合う列の入れ換えは奇数回になる
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2(交代性)から,行列式の変形に関する次の性質が導かれる.
2.1 2つの列が等しければ行列式の値は0になる.
2.2 1つの列の定数倍を他の列に加えても行列式の値は変わらない. 2.1 ← 2(交代性) の証明 例えば,1列目と2列目が等しいとき,これらを入れ換えると符号が変わるから したがって 2.2 ← 2.1 の証明 例えば,1列目に2列目の定数倍を加える場合 まず,線形性により2つに分けられる さらに,線形性により第2項の定数を前に出せる 2.1 により,2つの列が等しければ行列式は0になるから 以上により,他の列の定数倍を加えても行列式の値は変わらない. |
【行列式の変形規則:ここまでの要点】
1(多重線形性)・・・どの列に対しても線形である
【例】
2(交代性)・・・2つの列を入れ換えると符号が変わる
【例】
(2.1) 2つの列が等しければ行列式の値は0になる.
【例】
(2.2) 1つの列の定数倍を他の列に加えても行列式の値は変わらない.
【例】
※以上の性質は,行についても成り立つ. |
【クラメールの公式】・・・未知数が3個の連立1次方程式の場合
(証明)![]() dx+ey+fz=q …(*) gx+hy+iz=r の解は(分母が0でないとき)
※xの分子は,係数行列の1列目を右辺の列に入れ換えたもの.
yの分子は,係数行列の2列目を右辺の列に入れ換えたもの. zの分子は,係数行列の3列目を右辺の列に入れ換えたもの. とおくと,連立方程式(*)は次の形に書ける. ここで係数行列の行列式 において,1列目を(**)の左辺に書き換えると この行列式は1を使って次のように変形できる. さらに2.1を使うと,2つの列が等しい行列式の値は0になるから (***)は(**)により次の形に書ける 結局 2列目,3列目を書き換えると ■証明終わり■
【例題1】
(解答)次の連立方程式をクラメールの公式を使って解いてください. ![]() 3x+y−z=2 2x+y+3z=−2 係数行列の行列式は 係数行列の1列目,2列目,3列目を各々右辺の係数に入れ換えると
【例題2】
(解答)![]() x+y+z=1 2x−2y+z=4 次の連立方程式の解をクラメールの公式を使って求めてください. (暗算では無理です.各自計算用紙で求めてから下の選択肢のうちで正しいものをクリック)
【問題1】
![]() 3x+5y=7
【問題2】
![]() x−2y=−1
【問題3】
![]() x+2y+z=3 −x+2y+z=1
【問題4】
![]() 3x+2y−z=−1 x+4y−5z=−7
【問題5】
![]() 3x+z+2w=0 2x+y−w=2 x−y+3z+w=4 |
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■[個別の頁からの質問に対する回答][行列式の性質とクラメールの公式について/18.9.16]
この項で2つわからないところがありましたので申し訳ないですが質問をさせていただきます
質問1
この項のはじめにのところの、未知数が2この連立1次方程式の場合のところで、x,yの解がx=(pd-bq)/(ad-bc)、y=(aq-pc)/(ad-bc)
となっていますが
この項の1つ前の項の、未知数がn個で逆行列を用いて解く方法の項の例1のように、逆行列をかけてx,yをもとめると
x=dp-bq,y=aq-cpになりました
どうやったら、x,yの解がx=(pd-bq)/(ad-bc)、y=(aq-pc)/(ad-bc)になるんでしょうか?
質問2
この項の中下部にある、クラメールの公式と青点線でかこまれたところの、下にある、クラメールの公式の証明の上の方に、1列目を(**)の左辺にかきかえると、という文言がありますが、なぜ1列目を(**)の左辺にかきかえられるのでしょうか?
■[個別の頁からの質問に対する回答][行列式の性質とクラメールの公式について/17.5.18]
=>[作者]:連絡ありがとう.質問1:あなたは,逆行列の計算を間違っています. 質問2:行列式は方程式とは違います.関数や多項式のように値を持った式です.たとえば,f(x)=ax+bのときにxの代わりにpを代入するとf(p)=ap+bになりますが,この場面で「なぜ代入できるのか」という質問は,pは定義域にあるのかという質問になり,あなたの質問とは全然違う話になります. そもそも,そこの証明は,元の行列式をDとし,1列目を(**)の左辺に入れ換えたときの行列式をD’とすると,D’=xDになると書いているのです.(D’=Dだと書いているのではない.)だから,x=D’/D ←これを証明しています. クラメールの公式の問題1で下の式が間違っている
3x+5=7ではなく、3x+5y=7では?
=>[作者]:連絡ありがとう.問題文でyが抜けているということで,訂正しました. |
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