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--- lecture_15 [2016/03/10 11:36] – 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 50: / Line 61: @@
 \\&=-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 ====
+{{page>adjoint}}
+=== Example: $n=2$ ===
+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 $J=C^T=\mat{4&-2\\-3&1}$.