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--- lecture_15 [2015/03/24 10:22] – rupert
+++ lecture_15 [2017/03/28 11:16] (current) – [Definition: the adjoint of a square matrix] rupert
@@ Line 1: / Line 1: @@
-==== Step 4: the determinant of an $n\times n$ matrix ====
+==== Theorem: row/column operations and determinants ====
+Let $A$ be an $n\times n$ matrix, let $c$ be a scalar and let $i\ne j$.
-===Definition===
+$A_{Ri\to x}$ means $A$ but with row $i$ replaced by $x$.
-{{page>determinant of an nxn matrix}}
+  - If $i\ne j$, then $\det(A_{Ri\leftrightarrow Rj})=-\det(A)$ (swapping two rows changes the sign of det).
+  - $\det(A_{Ri\to c Ri}) = c\det(A)$ (scaling one row scales $\det(A)$ in the same way)
+  - $\det(A_{Ri\to Ri + c Rj}) = \det(A)$ (adding a multiple of one row to another row doesn't change $\det(A)$)
-===Example===
+  * Also, these properties all hold if you change "row" into "column" throughout.
-\begin{align*}
+==== Corollary ====
-\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}
-\vm{\color{red}1&\color{red}0&\color{red}2&\color{red}3\\0&2&1&-1\\2&0&0&1\\3&0&4&2} &= \color{red}1\vm{\color{blue}2&\color{blue}1&\color{blue}-1\\0&0&1\\0&4&2}-\color{red}0\vm{0&1&-1\\2&0&1\\3&4&2}+\color{red}2\vm{\color{orange}0&\color{orange}2&\color{orange}{-1}\\2&0&1\\3&0&2}-\color{red}3\vm{\color{purple}0&\color{purple}2&\color{purple}1\\2&0&0\\3&0&4}\\
-&= 1\left(\color{blue}2\vm{0&1\\4&2}-\color{blue}1\vm{0&1\\0&2}\color{blue}{-1}\vm{0&0\\0&4}\right)-0+2\left(\color{orange}0-\color{orange}{2}\vm{2&1\\3&2}\color{orange}{-1}\vm{2&0\\3&0}\right)-3\left(\color{purple}0-\color{purple}2\vm{2&0\\3&4}+\color{purple}1\vm{2&0\\3&0}\right)\\
-&=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)\\
-&=-8-4+48\\
-&=36.
-\end{align*}
-==== Theorem: Laplace expansion along any row or column gives the determinant ====
+If an $n\times n$ matrix $A$ has two equal rows (or columns), then $\det(A)=0$, and $A$ is not invertible.
-  - For any fixed $i$: $\det(A)=a_{i1}C_{i1}+a_{i2}C_{i2}+\dots+a_{in}C_{in}$ (Laplace expansion along row $i$)
+=== Proof ===
-  - For any fixed $j$: $\det(A)=a_{1j}C_{1j}+a_{2j}C_{2j}+\dots+a_{nj}C_{nj}$ (Laplace expansion along column $j$)
-\begin{align*}
-\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}
+If $A$ has two equal rows, row $i$ and row $j$, then $A=A_{Ri\leftrightarrow Rj}$
-\vm{1&0&2&3\\0&2&1&-1\\2&0&0&1\\3&0&4&2} &= 1\vm{2&1&-1\\0&0&1\\0&4&2}-0\vm{0&1&-1\\2&0&1\\3&4&2}+2\vm{0&2&-1\\2&0&1\\3&0&2}-3\vm{0&2&1\\2&0&0\\3&0&4}\\
+So $\det(A)=\det(A_{Ri\leftrightarrow Rj}) = -\det(A)$, so $2\det(A)=0$, so $\det(A)=0$.
-&= 1\left(2\vm{0&1\\4&2}-1\vm{0&1\\0&2}-1\vm{0&0\\0&4}\right)-0+2\left(0-2\vm{2&1\\3&2}-1\vm{2&0\\3&0}\right)-3\left(0-2\vm{2&0\\3&4}+1\vm{2&0\\3&0}\right)\\
-&=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)\\
+If $A$ has two equal columns, then $A^T$ has two equal rows, so $\det(A)=\det(A^T)=0$.
-&=-8-4+48\\
-&=36.
+In either case, $\det(A)=0$. So $A$ is not invertible.■
+=== Examples ===
+  * Swapping two rows changes the sign, so $\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\vm{0&0&2\\0&3&0\\4&0&0} = -\vm{4&0&0\\0&3&0\\0&0&2}=-4\cdot 3\cdot 2 = -24$.
+  * Multiplying a row or a column by a constant multiplies the determinant by that constant, so \begin{align*}\vm{ 2&4&6&10\\5&0&0&-10\\9&0&81&99\\1&2&3&4} &= 2\vm{ 1&2&3&5\\5&0&0&-10\\9&0&81&99\\1&2&3&4} \\&= 2\cdot 5\vm{ 1&2&3&5\\1&0&0&-2\\9&0&81&99\\1&2&3&4}\\&=  2\cdot 5\cdot 9 \vm{ 1&2&3&5\\1&0&0&-2\\1&0&9&11\\1&2&3&4}\\&=2\cdot 5\cdot 9\cdot 2 \vm{ 1&1&3&5\\1&0&0&-2\\1&0&9&11\\1&1&3&4}\\&=2\cdot 5\cdot 9\cdot 2\cdot 3 \vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4}.\end{align*}
+  * $\det(A_{R1\to R1-R4})=\det(A)$, so \begin{align*}\vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4} &=\vm{ 0&0&0&1\\1&0&0&-2\\1&0&3&11\\1&1&1&4}=-1\vm{1&0&0\\1&0&3\\1&1&1}+0\\&=-\vm{0&3\\1&1} = -(-3)=3.\end{align*}
+  * Hence \begin{align*}\vm{ 2&4&6&10\\5&0&0&-10\\9&0&81&99\\1&2&3&4} &= 2\cdot 5\cdot 9\cdot 2\cdot 3 \vm{ 1&1&1&5\\1&0&0&-2\\1&0&3&11\\1&1&1&4} \\&= 2\cdot 5\cdot 9\cdot 2\cdot 3 \cdot 3 = 1620.\end{align*}
+==== Corollary ===
+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 $\det(A)=0$.
+=== Proof ===
+Note that $\row_i(A)-c \cdot\row_j(A)=0$. So $A_{Ri\to Ri-c\,Rj}$ has a zero row, and by Laplace expansion along this row we obtain $\det(A_{Ri\to Ri-c\,Rj})=0$.  So $\det(A)=\det(A_{Ri\to Ri-c\,Rj})=0$.■
+==== The effect of EROs on the determinant ====
+We have now seen the effect of each of the three types of [[ERO]] on the determinant of a matrix:
+  - swapping two rows of the matrix multiplies the determinant by $-1$. By swapping rows repeatedly, we are able to shuffle the rows in an arbitrary fashion, and the determinant will either remain unchanged (if we used an even number of swaps) or be multiplied by $-1$ (if we used an odd number of swaps).
+  - multiplying one of the rows of the matrix by $c\in \mathbb{R}$ multiplies the determinant by $c$; and
+  - 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.
+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*}
+===== Finding the inverse of an invertible $n\times n$ matrix =====
+==== Definition: the adjoint of a square matrix ====
-=== Example ===
+{{page>adjoint}}
-=== Theorem ===
+=== Example: $n=2$ ===
-Let $A$ be an $n\times n$ matrix.
+If $A=\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}\def\vm#1{\begin{vmatrix}#1\end{vmatrix}}\mat{1&2\\3&4}$, then $C_{11}=+4$, $C_{12}=-3$, $C_{21}=-2$, $C_{22}=+1$. So the matrix of cofactors is $C=\mat{4&-3\\-2&1}$, so the adjoint of $A$ is $J=C^T=\mat{4&-2\\-3&1}$.
-  - $A$ is invertible if and only if $\det(A)\ne0$.
-  - If $A'$ is the same as $A$, except with two rows swapped, then $\det(A')=-\det(A)$.
-  - If $c$ is a scalar and $A'$ is the same as $A$ except with one row multiplied by $c$, then $\det(A')=c\det(A)$.
-  - If $A'$ and $A''$ are the same as $A$ except in row $i$, and $row_i(A'')=row_i(A)+row_i(A')$, then $\det(A'')=\det(A)+\det(A')$.
-  - $\det(A^T)=\det(A)$. So we can swap "row" with "column" in these properties.
-  - If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$.