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Determinanter (Kap. 6).md

@@ -25,53 +25,12 @@ $$\begin{aligned}\det(A)&=a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}&-a_{12}a_{21}a_{
 *Vi vet att* $$\det(A)=\sum^{m}_{j=n}(-1)^{i+j}a_{ij}\det(A_ij)\left(\text{Radutväkling med avsende på rad $i$}\right)$$
 **SATS**: *Låt $A$ vara en $m±times{n}$ diaonal matris. Då gäller det att* $$\begin{aligned}\det(A)=\prod^{m}_{i=1}a_{ii}\\A=\begin{bmatrix}a_{11}&0&0&0\\0&a_{22}&0&0\\0&0&a_{33}&0\\0&0&0&a_{44}\end{bmatrix}\end{aligned}$$
 - **BEVIS**: $$\begin{aligned}\text{(tänk på 4x4 exemplet) Om vi radutvklar med avsende på rad $1$ ges:}\\\det(A)=\sum^{4}_{j=1}(-1)^{1+j}\underset{\text{Den enda termen som inte är $0$ är $a_{11}$}}{a_{1j}}\det(A_{1j})=a_{11}\times\det(A_{11})=\\a_{11}\times\det\left(\begin{bmatrix}a_{22}&0&0\\0&a_{33}&0\\0&0&a_{44}\end{bmatrix}\right)\Rightarrow\\\text{$m$ raduväklar igen, med avsende på rad $1$ i den nya mindre matrisen:}\\=a_{11}\times{a_{22}}\times\det\left(\begin{bmatrix}a_{33}&0\\0&a_44\end{bmatrix}\right)=a_{11}\times{a_{22}}\times{a_{33}}\times\det(\begin{bmatrix}a_{44}\end{bmatrix})\end{aligned}$$
-	- **OBS**: *Samma resultat gäller för både över- ohc under-triangul'ra matriser:* $$
-\begin{aligned}
-\det\left(\begin{bmatrix}
-a_{11}&0&0&0\\
-a_{21}&a_{22}&0&0\\
-a_{31}&a_{32}&a_{33}&0\\
-a_{41}&a_{42}&a_{43}&a_{44}\\
-\end{bmatrix}\right)=a_{11}\times{a_{22}}\times{a_{33}}\times{a_{44}}\\
-\det\left(\begin{bmatrix}
-a_{11}&a_{12}&a_{13}&a_{14}\\
-0&a_{22}&a_{23}&a_{24}\\
-0&0&a_{33}&a_{34}\\
-0&0&0&a_{44}\\
-\end{bmatrix}\right)=a_{11}\times{a_{22}}\times{a_{33}}\times{a_{44}}
-\end{aligned}
-$$
+	- **OBS**: *Samma resultat gäller för både över- ohc under-triangul'ra matriser:* $$\begin{aligned}\det\left(\begin{bmatrix}a_{11}&0&0&0\\a_{21}&a_{22}&0&0\\a_{31}&a_{32}&a_{33}&0\\a_{41}&a_{42}&a_{43}&a_{44}\\\end{bmatrix}\right)=a_{11}\times{a_{22}}\times{a_{33}}\times{a_{44}}\\\det\left(\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}\\\end{bmatrix}\right)=a_{11}\times{a_{22}}\times{a_{33}}\times{a_{44}}\end{aligned}$$
 **SATS**: *Låt $A$ vara en $m\times{n}$ matris, och $\alpha\in\mathbb{R}$. Då gäller det att* $$\det(\alpha{A})=\underbracket{\alpha}\det(A)$$
-- **BEVIS**: $$\begin{aligned}
-\text{Kolla först $2\times2$ matriser: } A=\begin{bmatrix}
-a_{11}&a_{12}\\
-a_{21}&a_{22}
-\end{bmatrix}\Rightarrow\alpha{A}=\begin{bmatrix}
-\alpha a_{11}&\alpha a_{12}\\
-\alpha a_{21}&\alpha a_{22}
-\end{bmatrix}\\
-\text{Då gäller det att: }\det\left(\begin{bmatrix}
-\alpha a_{11}&\alpha a_{12}\\
-\alpha a_{21}&\alpha a_{22}
-\end{bmatrix}\right)=(\alpha{a_{11}})\times(\alpha{a_{22}})-(\alpha{a_{12}})\times(\alpha{a_{21}})=\\
-a^2(a_{11}a_{22}-a_{12}a_{21})=a^2\det(A)\\
-\text{För störe matriser följer resultater ur radutväklingsformel}
-\end{aligned}$$
+- **BEVIS**: $$\begin{aligned}\text{Kolla först $2\times2$ matriser: } A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\Rightarrow\alpha{A}=\begin{bmatrix}\alpha a_{11}&\alpha a_{12}\\\alpha a_{21}&\alpha a_{22}\end{bmatrix}\\\text{Då gäller det att: }\det\left(\begin{bmatrix}\alpha a_{11}&\alpha a_{12}\\\alpha a_{21}&\alpha a_{22}\end{bmatrix}\right)=(\alpha{a_{11}})\times(\alpha{a_{22}})-(\alpha{a_{12}})\times(\alpha{a_{21}})=\\a^2(a_{11}a_{22}-a_{12}a_{21})=a^2\det(A)\\\text{För störe matriser följer resultater ur radutväklingsformel}\end{aligned}$$
 **SATS** $$\begin{aligned}\text{Låt $A,B$ vara två $m\times{n}$ matriser. Då gäller det att}\\\det(AB)=\det(A)\times\det(B)\end{aligned}$$
-- **BEVIS** $$\begin{aligned}
-\text{Endast $2\times2$ matriser: }\\
-A=\begin{bmatrix}
-a_{11}&a_{12}\\
-a_{21}&a_{22}
-\end{bmatrix},\;B=\begin{bmatrix}
-b_{11}&b_{12}\\
-b_{21}&b_{22}
-\end{bmatrix},\;AB=\begin{bmatrix}
-a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\
-a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22}\\
-\end{bmatrix}\\
-\Rightarrow\det(AB)=(a_{11}b_{11}+a_{12}b_{21})\times(a_{11}b_{12}+a_{12}b_{22})\\-(a_{21}b_{11}+a_{22}b_{21})\times(a_{21}b_{12}+a_{22}b_{22})\\
-=(\cancel{a_{11}b_{11}a_{21}b_{12}}+a_{11}b_{11}a_{22}b_{22}+a_{11}b_{11}a_{22}b_{22}+\cancel{a_{12}b_{21}a_{22}b_{22}})\\-(\cancel{a_{11}b_{11}a_{21}b_{12}}+a_{11}b_{12}a_{22}b_{22}+a_{12}b_{22}a_{21}b_{11}+\cancel{a_{12}b_{21}a_{22}b_{22}})\\
-=a_{11}b_{11}a_{22}b_{22}+a_{11}b_{11}a_{22}b_{22}-a_{11}b_{12}a_{22}b_{22}-a_{12}b_{22}a_{21}b_{11}\\
-=a_{11}a_{22}(b_{11}b_{22}-b_{12}b_{21})-a_{12}a_{21}(b_{11}b_{22}-b_{12}b_{21})
-\end{aligned}$$
+- **BEVIS** $$\begin{aligned}\text{Endast $2\times2$ matriser: }\\A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix},\;B=\begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix},\;AB=\begin{bmatrix}a_{11}b_{11}+a_{12}b_{21}&a_{11}b_{12}+a_{12}b_{22}\\a_{21}b_{11}+a_{22}b_{21}&a_{21}b_{12}+a_{22}b_{22}\\\end{bmatrix}\\\Rightarrow\det(AB)=(a_{11}b_{11}+a_{12}b_{21})\times(a_{11}b_{12}+a_{12}b_{22})\\-(a_{21}b_{11}+a_{22}b_{21})\times(a_{21}b_{12}+a_{22}b_{22})\\=(\cancel{a_{11}b_{11}a_{21}b_{12}}+a_{11}b_{11}a_{22}b_{22}+a_{11}b_{11}a_{22}b_{22}+\cancel{a_{12}b_{21}a_{22}b_{22}})\\-(\cancel{a_{11}b_{11}a_{21}b_{12}}+a_{11}b_{12}a_{22}b_{22}+a_{12}b_{22}a_{21}b_{11}+\cancel{a_{12}b_{21}a_{22}b_{22}})\\=a_{11}b_{11}a_{22}b_{22}+a_{11}b_{11}a_{22}b_{22}-a_{11}b_{12}a_{22}b_{22}-a_{12}b_{22}a_{21}b_{11}\\=a_{11}a_{22}(b_{11}b_{22}-b_{12}b_{21})-a_{12}a_{21}(b_{11}b_{22}-b_{12}b_{21})\\=a_{11}a_{22}\times\det(B)-a_{12}a_{21}\times\det(B)=(a_{11}a_{22}-a_{12}a_{21})\det(B)=\det(A)\det(B)\end{aligned}$$
+**SATS**: *Låt $A$ vata en $m\times{n}$ matris. Då gäller: *$$\det(A)=\det(A^T)$$
+- **BEVIS**: $$\begin{aligned}\text{Endast $2\times2$: }\left.\begin{aligned}A=\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\Rightarrow\det(A)=a_{11}a_{22}-a_{12}a_{21}\\A^T=\begin{bmatrix}a_{11}&a_{21}\\a_{12}&a_{22}\end{bmatrix}\Rightarrow\det(A)=a_{11}a_{22}-a_{21}a_{12}\end{aligned}\right\}\text{Exakt samma}\end{aligned}$$
+**DEF**: *Låt $A$ vara $m\times{n}$ matris. Denna matrisen kofaktormatris är den $m\times{n}$ matrisen $\operatorname{cof}(A)$ vars element i rad $i$ och kolumn $j$ är *$$\begin{aligned}(-1)^{1+j}\det(A_{ij})\end{aligned}$$
+- **EX**: $$\begin{aligned}A=\begin{bmatrix}1&1&6\\-3&-4&-16\\3&5&13\end{bmatrix}\Rightarrow\operatorname{cof}(A)=\begin{bmatrix}+\begin{vmatrix}-4&-16\\5&13\end{vmatrix}&-\begin{vmatrix}-3&-16\\3&13\end{vmatrix}&+\begin{vmatrix}-3&-4\\3&5\end{vmatrix}\\-\begin{vmatrix}1&6\\5&13\end{vmatrix}&+\begin{vmatrix}1&6\\3&16\end{vmatrix}&-\begin{vmatrix}1&1\\3&5\end{vmatrix}\\+\begin{vmatrix}1&6\\-4&-16\end{vmatrix}&-\begin{vmatrix}1&6\\-3&-16\end{vmatrix}&+\begin{vmatrix}1&1\\-3&-4\end{vmatrix}\end{bmatrix}\\=\begin{bmatrix}28&-9&-3\\17&-5&-2\\8&-2&-1\end{bmatrix}\end{aligned}$$