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--- lecture_15 [2016/03/29 13:51] – [Definition: the adjoint of a square matrix] rupert
+++ lecture_15 [2017/03/28 11:16] (current) – [Definition: the adjoint of a square matrix] rupert
@@ Line 1: / Line 1: @@
+==== Theorem: row/column operations and determinants ====
+Let $A$ be an $n\times n$ matrix, let $c$ be a scalar and let $i\ne j$.
+$A_{Ri\to x}$ means $A$ but with row $i$ replaced by $x$.
+  - 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 $\det(A)$)
+  * Also, these properties all hold if you change "row" into "column" throughout.
 ==== Corollary ====
@@ Line 59: / Line 70: @@
 === Example: $n=2$ ===
-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 $J=C^T=\mat{d&-b\\-c&a}$.
+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 $J=C^T=\mat{4&-2\\-3&1}$.
-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 $A^{-1}=\frac1{\det A}J$.
-=== Example: $n=3$ ===
-If $A=\mat{3&1&0\\-2&-4&3\\5&4&-2}$, then the matrix of signs is $\mat{+&-&+\\-&+&-\\+&-&+}$, so
-\[ C=\mat{
-\vm{-4&3\\4&-2}&-\vm{-2&3\\5&-2}&\vm{-2&-4\\5&4}\\
--\vm{1&0\\4&-2}&\vm{3&0\\5&-2}&-\vm{3&1\\5&4}\\
-\vm{1&0\\-4&3}&-\vm{3&0\\-2&3}&\vm{3&1\\-2&-4}}
-= \mat{-4&11&12\\2&-6&-7\\3&-9&-10}\]
-so the adjoint of $A$ is
-\[ J=C^T=\mat{-4&2&3\\11&-6&-9\\12&-7&-10}.\]