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--- lecture_13 [2016/03/07 10:49] – rupert
+++ lecture_13 [2017/03/07 15:08] (current) – [Step 3: the determinant of a $3\times 3$ matrix using Laplace expansion along the first row] rupert
@@ Line 1: / Line 1: @@
+==== Example ====
+Let's solve the matrix equation $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}\mat{1&5\\3&-2}X=\mat{4&1&0\\0&2&1}$ for $X$.
+Write $A=\mat{1&5\\3&-2}$. Then $\det(A)=1(-2)-5(3)=-2-15=-17$ which isn't zero, so $A$ is invertible. And $A^{-1}=\frac1{-17}\mat{-2&-5\\-3&1}=\frac1{17}\mat{2&5\\3&-1}$.
+Hence the solution is $X=A^{-1}\mat{4&1&0\\0&2&1}=\frac1{17}\mat{2&5\\3&-1}\mat{4&1&0\\0&2&1}=\frac1{17}\mat{8&12&5\\12&1&-1}$.
 ====== The transpose of a matrix ======
@@ Line 18: / Line 27: @@
 If $a$ is a $1\times m$ row vector and $b$ is an $m\times 1$ column vector, then
 \[ ab=b^Ta^T.\]
+==== Observation: the transpose swaps rows with columns ====
+Formally, for any matrix $A$ and any $i,j$, we have
+\begin{align*}\def\col#1{\text{col}_{#1}}\def\row#1{\text{row}_{#1}}
+\row i(A^T)&=\col i(A)^T\\\col j(A^T)&=\row j(A)^T
+.\end{align*}
+==== Theorem: the transpose reverses the order of matrix multiplication ====
+If $A$ and $B$ are matrices and the [[matrix product]] $AB$ is defined, then $B^TA^T$ is also defined. Moreover, in this case we have
+\[ (AB)^T=B^TA^T.\]
+=== Proof ===
+If $AB$ is defined, then $A$ is $n\times m$ and $B$ is $m\times k$ for some $n,m,k$, so $B^T$ is $k\times m$ and $A^T$ is $m\times n$, so $B^TA^T$ is defined. Moreover, in this case $B^TA^T$ is an $k\times n$ matrix, and $AB$ is an $n\times k$ matrix, so $(AB)^T$ is a $k\times n$ matrix. Hence $B^TA^T$ has the same size as $(AB)^T$. To show that they are equal, we calculate, using the fact that the transpose swaps rows with columns:
+\begin{align*}
+\text{the }(i,j)\text{ entry of }(AB)^T&= \text{the }(j,i)\text{ entry of }AB
+\\&= \row j(A)\cdot\col i(B)
+\\&=\col i(B)^T\cdot \row j(A)^T \quad\text{by the previous Lemma}
+\\&=\row i(B^T)\cdot \col j(A^T) \quad\text{by the Observation}
+\\&=\text{the }(i,j)\text{ entry of }B^TA^T
+\end{align*}
+Hence $(AB)^T=B^TA^T$. ■
+====== Determinants of $n\times n$ matrices ======
+Given any $n\times n$ matrix $A$, it is possible to define a number $\det(A)$  (as a formula using the entries of $A$) so that
+\[ A\text{ is invertible} \iff \det(A)\ne0.\]
+  - If $A$ is a $1\times 1$ matrix, say $A=[a]$, then we just define $\det[a]=a$.
+  - If $A$ is a $2\times 2$ matrix, say $A=\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}\mat{a&b\\c&d}$, then we've seen that $\det(A)=ad-bc$.
+  - If $A$ is a $3\times 3$ matrix, say $A=\mat{a&b&c\\d&e&f\\g&h&i}$, then it turns out that $\det(A)=aei-afh+bfg-bdi+cdh-ceg$.
+  - If $A$ is a $4\times 4$ matrix, then the formula for $\det(A)$ is more complicated still, with $24$ terms.
+  - If $A$ is a $5\times 5$ matrix, then the formula for $\det(A)$ has $120$ terms.
+Trying to memorise a formula in every case (or even in the $3\times 3$ case!) isn't convenient unless we understand it somehow. We will approach this is several steps.
+==== Step 1: minors ====
+=== Definition ===
+{{page>minor}}
+=== Examples ===
+  * If $A=\mat{3&5\\-4&7}$, then $M_{11}=\det[7]=7$, $M_{12}=\det[-4]=-4$, $M_{21}=5$, and $M_{22}=3$.
+  * If $A=\mat{1&2&3\\7&8&9\\11&12&13}$, then $M_{23}=\det\mat{1&2\\11&12}=1\cdot 12-2\cdot 11=-10$ and $M_{32}=\det\mat{1&3\\7&9}=-12$.
+==== Step 2: cofactors ====
+===Definition===
+{{page>cofactor}}
+Note that $(-1)^{i+j}$ is $+1$ or $-1$, and can looked up in the matrix of signs:
+$\mat{+&-&+&-&\dots\\-&+&-&+&\dots\\+&-&+&-&\dots\\\vdots&\vdots&\vdots&\vdots&\ddots}$.
+This matrix starts with a $+$ in the $(1,1)$ entry (corresponding to $(-1)^{1+1}=(-1)^2=+1$) and the signs then alternate.
+===Examples===
+  * If $A=\mat{3&5\\-4&7}$, then $C_{11}=+M_{11}=\det[7]=7$, $C_{12}=-M_{12}=-\det[-4]=4$, $C_{21}=-5$, and $C_{22}=3$.
+  * If $A=\mat{1&2&3\\7&8&9\\11&12&13}$, then $C_{23}=-M_{23}=-(-10)=10$ and $C_{33}=+M_{33}=\det\mat{1&2\\7&8}=-6$.
+==== Step 3: the determinant of a $3\times 3$ matrix using Laplace expansion along the first row ====
+===Definition===
+{{page>determinant of a 3x3 matrix}}