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Teď si ukážeme můj oblíbený způsob hledání

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inverzní matice 3x3

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vlastně si myslím, že to je mnohem zábavnější způsob.

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je také pravděpodobné, že takto uděláte méně chyb

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Pokud si dobře vzpomínám na Algebru 2

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tak tam tento způsob neukazovali.

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Proto jsem nejdříve ukázal jiný způsob.

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Pojďme se na to podívat.

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A v budoucnu vám vysvětlím, proč to funguje

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To je vždy důležité.

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Toto je ale část lineární algebry,

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u které je důležité naučit se nejdříve postup

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a později se naučit princip.

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protože je to velmi mechanické

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a v podstatě obsahuje pouze

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jednoduchou aritmetiku

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Ale vysvětlení je mnohem složitější,

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takže to nechám na další videa.

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Můžete často začít přemýšlet do hloubky

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když máte dojem, že víte, jak na to.

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Pojďme se podívat na naší původní matici

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Jakou matici jsme řešili

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v posledním videu?

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byla to tato

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a chtěli jsme k ní najít inverzní matici

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Tak se do toho pustíme.

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Tento postup hledání inverzní matice

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se nazývá Gauss-Jordanova eliminace.

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Způsob, jakým se to dělá,

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může vypadat jako magie,

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ale z dalších videí pochopíte, že to dává smysl

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Rozšíříme tuto matici

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Co znamená "rozšířit"?

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Něco přidat.

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takže udělám dělící čáru

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někdo to tak nedělá.

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Takže, moje dělící čára

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Co "přidáme" na druhou stranu?

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přidáme identickou matici stejného rozměru

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V případě 3x3 to bude jedn. matice 3x3

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to je takto.

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Co uděláme teď?

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Provedu teď řadu

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"ekvivalentních řádkových úprav"

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Vysvětlím, co jsou "ekv. řádk. úpravy"

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této matice.

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Důležité je, že jakékoli úpravy dělám zde

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musím provést u celé odpovídající řády

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Cílem je provést sérii určitých operací

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na levé straně -

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samozřejmě ty samé provedeme i na pravé

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straně, takže nakonec na levé straně

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vytvořím jednotkovou matici.

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A když už mám nalevo jednotkovou matici,

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na pravé straně mám inverzní matici

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té původní matice. Tento tvar s

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jednotkovou maticí se

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nazývá "diagonální"

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Promluvím o tom více později.

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V linearní algebře máme mnoho jmen a defi-

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-nic, jsou to ale většinou

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jednoduché koncepty. Tak začneme

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a bude to hned jasnější

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minimálně početní postup, princip ovšem

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asi ne.

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Takže nejdříve se snažím, jak jsem řekl,

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udělat určité úpravy.

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Jaké jsou povolené úpravy?

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Říká se jim "ekvivalentní řádkové úpravy"

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Můžu provést několik operací

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Můžu zaměnit řadu

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jejím násobkem

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To je možné.

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Můžu prohodit jakékoliv dvě řady.

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Samozřejmě pokud prohodím

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dvě řady, provedu to i na pravé straně.

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Mohu sečíst nebo odečíst dvě řady

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Takže například zaměním tuto řadu

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s touto, ke které jsem přidal jinou řadu.

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Za chvilku uvidíte.

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Takže můžeme třeba

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vynásobit tuto řadu (-1), příčíst jí

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k této a napsat jí místo té původní.

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Pokud vám to připomíná něco jako

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když jste řešili soustavu lineárních

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rovnic, tak to není náhoda.

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Matice jsou totiž hodně dobrý způsob

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jak soustavy reprezentovat.

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proveďme teď určité EŘÚ

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,na levé straně provedeme úplnou eliminaci

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tzn. jinými slovy

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uděláme z ní jednotkovou matici

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Podívejme se na to, co tedy chceme udělat.

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tady všude chceme mít jedničky

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a tady nuly.

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Kouknem se teď jak to dělat efektivně

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Přepíšu si ještě jednou tu matici

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Tady si uděláme nuly

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to bude užitečné

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Takže, první dvě řady necháme být

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1,0,1

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tady je dělicí čára

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