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lecture_14_slides

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lecture_14_slides [2017/03/08 17:00] – [Definition: upper triangular matrices] rupertlecture_14_slides [2017/03/09 10:45] (current) – [Corollary: the determinant of an upper triangular matrix] rupert
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 Expand $\det(A)$ along the zero row or column Expand $\det(A)$ along the zero row or column
-  - get a sum of terms $0\times C_{ij}$ +  * get a sum of terms $0\times C_{ij}$ 
-  - so $\det(A)=0$ +  * so $\det(A)=0$ 
-  - so $A$ isn't invertible.+  * so $A$ isn't invertible.■
  
-==== Definition: upper triangular matrices ====+==== Definition: "upper triangular" and "diagonal" ====
  
-An $n\times n$ matrix $A$ is **upper triangular** if all the entries below the main diagonal are zero.+An $n\times n$ matrix is **upper triangular** if all the entries below the main diagonal are zero.
  
  
-==== Definition: diagonal matrices ==== +An $n\times n$ matrix is **diagonal** if the only non-zero entries are on its main diagonal.
- +
-An $n\times n$ matrix $A$ is **diagonal** if the only non-zero entries are on its main diagonal.+
  
 ==== Corollary: the determinant of an upper triangular matrix ==== ==== Corollary: the determinant of an upper triangular matrix ====
  
-The determinant of an upper triangular $n\times n$ matrix is the product of its diagonal entries: \[\det(A)=a_{11}a_{22}\dots a_{nn}.\]+If $A$ is an upper triangular matrix or a diagonal matrix, then the determinant of $A$ is the product of its diagonal entries: \[\det(A)=a_{11}a_{22}\dots a_{nn}.\] 
 + 
 +  * Every diagonal matrix is upper triangular! 
 +  * So it suffices just to prove it for upper triangular matrices.
  
 ==== Proof by induction on $n$ ==== ==== Proof by induction on $n$ ====
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   * So $\det(A)=a_{11}C_{11}=a_{11}a_{22}a_{33}\dots a_{nn}$.■   * So $\det(A)=a_{11}C_{11}=a_{11}a_{22}a_{33}\dots a_{nn}$.■
  
-==== Corollary: the determinant of a diagonal matrix ==== 
  
-The determinant of an $n\times n$ diagonal matrix is the product of its diagonal entries: \[\det(A)=a_{11}a_{22}\dots a_{nn}.\] 
- 
-=== Proof === 
-  * Any diagonal matrix is upper triangular, so this is a special case of the last Corollary (about upper triangular matrices).■ 
  
  
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   - If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$.   - If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$.
  
 +=== Corollary on invertibility ===
 +
 +  - $A^T$ is invertible if and only if $A$ is invertible
 +  - $AB$ is invertible if and only if **both** $A$ and $B$ are invertible
 +
 +
 +  * Warning: it's not true that $A+B$ is invertible if and only if $A$ and $B$ are invertible!
 ==== Theorem: row/column operations and determinants ==== ==== Theorem: row/column operations and determinants ====
  
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 ==== Corollary === ==== Corollary ===
  
-If $\def\row{\text{row}}\row_i(A)=c\cdot \row_j(A)$ for some $i\ne j$ and some $c\in \mathbb{R}$, then $\det(A)=0$.+If $\def\row{\text{row}}\row_i(A)=c\cdot \row_j(A)$ for some $i\ne j$ and some $c\in \mathbb{R}$, then $\det(A)=0$ (and so $A$ isn't invertible).
  
 === Proof === === Proof ===
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   * $\row_i(A)-c \cdot\row_j(A)=0$   * $\row_i(A)-c \cdot\row_j(A)=0$
   * So $A_{Ri\to Ri-c\,Rj}$ has a zero row   * So $A_{Ri\to Ri-c\,Rj}$ has a zero row
-  * By Laplace expansion along this row: $\det(A_{Ri\to Ri-c\,Rj})=0$+  * So $\det(A_{Ri\to Ri-c\,Rj})=0$
   * So $\det(A)=\det(A_{Ri\to Ri-c\,Rj})=0$.■   * So $\det(A)=\det(A_{Ri\to Ri-c\,Rj})=0$.■
  
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 \begin{align*}\vm{1&3&1&3\\\color{red}4&\color{red}8&\color{red}0&\color{red}{12}\\0&1&3&6\\2&2&1&6}&= \color{red}{4}\vm{1&3&1&\color{blue}3\\1&2&0&\color{blue}3\\0&1&3&\color{blue}6\\2&2&1&\color{blue}6}=4\cdot \color{blue}3\vm{\color{green}1&3&1&1\\\color{red}1&2&0&1\\\color{red}0&1&3&2\\\color{red}2&2&1&2} \begin{align*}\vm{1&3&1&3\\\color{red}4&\color{red}8&\color{red}0&\color{red}{12}\\0&1&3&6\\2&2&1&6}&= \color{red}{4}\vm{1&3&1&\color{blue}3\\1&2&0&\color{blue}3\\0&1&3&\color{blue}6\\2&2&1&\color{blue}6}=4\cdot \color{blue}3\vm{\color{green}1&3&1&1\\\color{red}1&2&0&1\\\color{red}0&1&3&2\\\color{red}2&2&1&2}
-=12\vm{1&3&1&1\\\color{blue}0&\color{blue}{-1}&\color{blue}{-1}&\color{blue}{0}\\\color{blue}0&\color{blue}1&\color{blue}3&\color{blue}2\\0&-4&-1&-0} +\\&=12\vm{1&3&1&1\\\color{blue}0&\color{blue}{-1}&\color{blue}{-1}&\color{blue}{0}\\\color{blue}0&\color{blue}1&\color{blue}3&\color{blue}2\\0&-4&-1&-0} 
-\\&=\color{blue}{-}12\vm{1&3&1&1\\0&\color{green}1&3&2\\0&\color{red}{-1}&{-1}&{0}\\0&\color{red}{-4}&-1&0} +=\color{blue}{-}12\vm{1&3&1&1\\0&\color{green}1&3&2\\0&\color{red}{-1}&{-1}&{0}\\0&\color{red}{-4}&-1&0} 
-=-12\vm{1&3&1&1\\0&1&3&2\\0&0&\color{green}2&2\\0&0&\color{red}{11}&8}+\\&=-12\vm{1&3&1&1\\0&1&3&2\\0&0&\color{green}2&2\\0&0&\color{red}{11}&8}
 =-12\vm{1&3&1&1\\0&1&3&2\\0&0&2&2\\0&0&0&-3} =-12\vm{1&3&1&1\\0&1&3&2\\0&0&2&2\\0&0&0&-3}
 \\&=-12(1)(1)(2)(-3)=72. \\&=-12(1)(1)(2)(-3)=72.
 \end{align*} \end{align*}
 +
  
  
  
lecture_14_slides.1488992457.txt.gz · Last modified: by rupert
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