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--- lecture_16 [2015/03/26 10:29] – rupert
+++ lecture_16 [2017/03/30 09:20] (current) – [Chapter 3: Vectors and geometry] rupert
@@ Line 1: / Line 1: @@
-==== Corollary ===
+=== Example: $n=2$, general case ===
-If $\def\row{\text{row}}\row_j(A)=c\cdot \row_i(A)$ for some $i\ne j$ and some $c\in \mathbb{R}$, then $\det(A)=0$.
+If $A=\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\mat{a&b\\c&d}$, then $C=\mat{d&-c\\-b&a}$, so the adjoint of $A$ is $J=C^T=\mat{d&-b\\-c&a}$.
-=== Proof ===
+Recall that $AJ=(\det A)I_2=JA$; we calculated this earlier when we looked at the inverse of a $2\times 2$ matrix. Hence for a $2\times 2$ matrix $A$, if $\det A\ne0$, then $A^{-1}=\frac1{\det A}J$.
-Let $B$ have the same rows as $A$, except with $\row_i(A)$ in both row $i$ and row $j$. Observe that
+=== Example: $n=3$ ===
-  * row $j$ of $B$ is $c$ times row $j$ of $A$, and all the other rows are equal; and
+If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{3&1&0\\-2&-4&3\\5&4&-2}$, then the matrix of signs is $\mat{+&-&+\\-&+&-\\+&-&+}$, so
-  * $A$ has two equal rows.
+\[\def\vm#1{\begin{vmatrix}#1\end{vmatrix}} C=\mat{
-Hence using property 3 in the theorem and the previous corollary, we have $\det(B)=c\cdot \det(A)=c\cdot 0=0.$ ■
+\vm{-4&3\\4&-2}&-\vm{-2&3\\5&-2}&\vm{-2&-4\\5&4}\\
+-\vm{1&0\\4&-2}&\vm{3&0\\5&-2}&-\vm{3&1\\5&4}\\
+\vm{1&0\\-4&3}&-\vm{3&0\\-2&3}&\vm{3&1\\-2&-4}}
+= \mat{-4&11&12\\2&-6&-7\\3&-9&-10}\]
+so the adjoint of $A$ is
+\[ J=C^T=\mat{-4&2&3\\11&-6&-9\\12&-7&-10}.\]
-==== Corollary ====
+Observe that $AJ=\mat{3&1&0\\-2&-4&3\\5&4&-2}\mat{-4&2&3\\11&-6&-9\\12&-7&-10}=\mat{-1&0&0\\0&-1&0\\0&0&-1}=-1\cdot I_3$, and $JA=\mat{-4&2&3\\11&-6&-9\\12&-7&-10}\mat{3&1&0\\-2&-4&3\\5&4&-2}=\mat{-1&0&0\\0&-1&0\\0&0&-1}=-1\cdot I_3$; and $\det(A)=-1$.
-If $E$ has the same rows as $A$ except in row $j$, and $\row_j(E)=\row_j(A)+c\cdot \row_i(A)$ for some $i\ne j$ and some $c\in \mathbb{R}$, then $\det(E)=\det(A)$.
+This is an illustration of the following theorem, whose proof is omitted:
+==== Theorem: key property of the adjoint of a square matrix ====
+If $A$ is any $n\times n$ matrix and $J$ is its [[adjoint]], then $AJ=(\det A)I_n=JA$.
+==== Corollary: a formula for the inverse of a square matrix ====
+If $A$ is any $n\times n$ matrix with $\det(A)\ne 0$, then $A$ is invertible and \[A^{-1}=\frac1{\det A}J\] where $J$ is the adjoint of $A$.
 === Proof ===
-Let $B$ have the same rows as $A$, except with $c\cdot \row_i(A)$ in row $j$. Observe that
+Divide the equation $AJ=(\det A)I_n=JA$ by $\det A$.  ■
+==== Example ====
+If again we take $A=\mat{3&1&0\\-2&-4&3\\5&4&-2}$, then $J=\mat{-4&2&3\\11&-6&-9\\12&-7&-10}$ and $\det(A)=-1$, so $A$ is invertible and $A^{-1}=\frac1{-1}J=-J=\mat{4&-2&-3\\-11&6&9\\-12&7&10}$.
+==== Example ($n=4$) ====
+Let $A=\mat{1&0&0&0\\1&2&0&0\\1&2&3&0\\1&2&3&4}$.
+Recall that a matrix with a repeated row or a zero row has determinant zero.
+We have \[C=\mat{+\vm{2&0&0\\2&3&0\\2&3&4}&-\vm{1&0&0\\1&3&0\\1&3&4}&+0&-0\\-0&+\vm{1&0&0\\1&3&0\\1&3&4}&-\vm{1&0&0\\1&2&0\\1&2&4}&+0\\+0&-0&+\vm{1&0&0\\1&2&0\\1&2&4}&-\vm{1&0&0\\1&2&0\\1&2&3}\\-0&+0&-0&+\vm{1&0&0\\1&2&0\\1&2&3}}=\mat{24&-12&0&0\\0&12&-8&0\\0&0&8&-6\\0&0&0&6}\]
+so \[J=C^T=\mat{24&0&0&0\\-12&12&0&0\\0&-8&8&0\\0&0&-6&6}.\]
+Since $A$ is lower triangular, its determinant is given by multiplying together its diagonal entries: $\det(A)=1\times 2\times 3\times 4=24$. (Note that even if $A$ was not triangular, $\det A$ can be easily found from the matrix of cofactors $C$ by summing the entries of $A$ multiplied by the entries of $C$ (i.e., the minors) along any row or column.)
+So \[A^{-1}=\frac1{\det A}J = \frac1{24}\mat{24&0&0&0\\-12&12&0&0\\0&-8&8&0\\0&0&-6&6}=\mat{1&0&0&0\\-1/2&1/2&0&0\\0&-1/3&1/3&0\\0&0&-1/4&1/4}.\]
+You should check that this really is the inverse, by checking that $AA^{-1}=I_4=A^{-1}A$.
+===== A more efficient way to find $A^{-1}$ =====
-  * $\det(B)=0$, by the previous corollary; and
+Given an $n\times n$ matrix $A$, form the $n\times 2n$ matrix
-  * except in row $j$, the rows of $E$, $A$ and $B$ are all equal, and $\row_j(E)=\row_j(A)+\row_j(B)$.
+\[ \def\m#1{\left[
+\begin{array}{@{} c|c {}@} % it does autodetection
+#1
+\end{array}
+\right]}\m{A&I_n}\]
+and use [[EROs]] to put this matrix into [[RREF]]. One of two things can happen:
-Hence by property 4 of the theorem, we have $\det(E)=\det(A)+\det(B)=\det(A)+0=\det(A)$. ■
+  * Either you get a row of the form $[0~0~\dots~0~|~*~*~\dots~*]$ which starts with $n$ zeros. You can then conclude that $A$ is not invertible.
+  * Or you end up with a matrix of the form $\m{I_n&B}$ for some $n\times n$ matrix $B$. You can then conclude that $A$ is invertible, and $A^{-1}=B$.
-We have now seen the effect of each of the three types of [[ERO]] on the determinant of a matrix:
+==== Examples ====
-  - changing the order of the rows of the matrix multiplies the determinant by $-1$;
+  * Consider $A=\def\mat#1{\begin{matrix}#1\end{matrix}}\left[\mat{1&3\\2&6}\right]$. \begin{align*}\m{A&I_2}&=\m{\mat{1&3\\2&6}&\mat{1&0\\0&1}}
-  - multiplying one of the rows of the matrix by $c\in \mathbb{R}$ multiplies the determinant by $c$; and
+\def\go#1#2{\m{\mat{#1}&\mat{#2}}}
-  - replacing row $j$ by "row $j$ ${}+{}$ $c\times {}$ (row $i$)", where $c$ is a non-zero real number and $i\ne j$ does not change the determinant.
+\def\ar#1{\\[6pt]\xrightarrow{#1}&}
+\ar{R2\to R2-2R1}\go{1&3\\0&0}{1&0\\-2&1}
+\end{align*} Conclusion: $A$ is not invertible.
+  * Consider $A=\left[\mat{1&3\\2&7}\right]$.\begin{align*}\m{A&I_2}&=\m{\mat{1&3\\2&7}&\mat{1&0\\0&1}}
+\ar{R2\to R2-2R1}\go{1&3\\0&1}{1&0\\-2&1}
+\ar{R1\to R1-3R1}\go{1&0\\0&1}{7&-3\\-2&1}
+\end{align*} Conclusion: $A$ is invertible and $A^{-1}=\left[\mat{7&-3\\-2&1}\right]$.
+  * Consider $A=\left[\mat{3&1&0\\-2&-4&3\\5&4&-2}\right]$.\begin{align*}\m{A&I_3}&=\go{3&1&0\\-2&-4&3\\5&4&-2}{1&0&0\\0&1&0\\0&0&1}
+\ar{R1\to R1+R2}
+\go{1&-3&3\\-2&-4&3\\5&4&-2}{1&1&0\\0&1&0\\0&0&1}
+\ar{R2\to R2+2R1,\ R3\to R3-5R1}
+\go{1&-3&3\\0&-10&9\\0&19&-17}{1&1&0\\2&3&0\\-5&-5&1}
+\ar{R3\leftrightarrow R2}
+\go{1&-3&3\\0&19&-17\\0&-10&9}{1&1&0\\-5&-5&1\\2&3&0}
+\ar{R2\to R2+2R3}
+\go{1&-3&3\\0&-1&1\\0&-10&9}{1&1&0\\-1&1&1\\2&3&0}
+\ar{R1\to R1+3R2,\ R3\to R3-10R2}
+\go{1&0&0\\0&-1&1\\0&0&-1}{4&-2&3\\-1&1&1\\12&-7&-10}
+\ar{R2\to R2+R3}
+\go{1&0&0\\0&-1&0\\0&0&-1}{4&-2&3\\11&-6&-9\\12&-7&-10}
+\ar{R2\to -R2,\ R3\to -R3}
+\go{1&0&0\\0&1&0\\0&0&1}{4&-2&3\\-11&6&9\\-12&7&10}
+\end{align*} Conclusion: $A$ is invertible, and $A^{-1}=\left[\mat{4&-2&3\\-11&6&9\\-12&7&10}\right]$.
-Moreover, since $\det(A)=\det(A^T)$, this all applies equally to columns instead of rows.
-We can use EROs to put a matrix into upper triangular form, and then finding the determinant is easy: just multiply the diagonal entries together. We just have to keep track of how the determinant is changed by the EROs of types 1 and 2.
-==== Example: using EROs to find the determinant ====
-\begin{align*}\def\vm#1{\begin{vmatrix}#1\end{vmatrix}} \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}
-\\&=\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&2&2\\0&0&0&-3}
-\\&=-12(1)(1)(2)(-3)=72.
-\end{align*}