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--- lecture_14 [2016/03/10 11:36] – rupert
+++ lecture_14 [2017/03/09 10:49] (current) – [Corollary on invertibility] rupert
@@ Line 1: / Line 1: @@
+=== Example ===
+\begin{align*}\def\mat#1{\left[\begin{smallmatrix}#1\end{smallmatrix}\right]}\det\mat{1&2&3\\7&8&9\\11&12&13} &= 1\cdot C_{11} + 2 C_{12} + 3 C_{13}\\
+&= 1 \cdot (+M_{11}) + 2 \cdot (-M_{12}) + 3 \cdot(+M_{13})\\
+&= M_{11}-2M_{12}+3M_{13}\\
+&= \det\mat{8&9\\12&13} -2\det\mat{7&9\\11&13} + 3\det\mat{7&8\\11&12}\\
+&= (8\cdot 13-9\cdot 12) -2(7\cdot 13-9\cdot 11)+3(7\cdot 12-8\cdot 11)\\
+&=-4 -2(-8)+3(-4)\\
+&=-4+16-12\\
+&=0.\end{align*}
+From this, we can conclude that $\mat{1&2&3\\7&8&9\\11&12&13}$ is //not// invertible.
 === Notation ===
@@ Line 77: / Line 90: @@
   - If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$
-==== Theorem: row/column operations and determinants ====
+==== Corollary on invertibility ====
-Let $A$ be an $n\times n$ matrix, let $c$ be a scalar and let $i\ne j$.
+  - $A^T$ is invertible if and only if $A$ is invertible
+  - $AB$ is invertible if and only if **both** $A$ and $B$ are invertible
-$A_{Ri\to x}$ means $A$ but with row $i$ replaced by $x$.
+=== Proof ===
-  - 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.
+  - We have $\det(A^T)=\det(A)$. So $A^T$ is invertible $\iff$ $\det(A^T)\ne0$ $\iff$ $\det(A)\ne 0$ $\iff$ $A$ is invertible.
+  - We have $\det(AB)=\det(A)\det(B)$. So $AB$ is invertible $\iff$ $\det(AB)\ne 0$ $\iff$ $\det(A)\det(B)\ne0$ $\iff$ $\det(A)\ne0$ and $\det(B)\ne 0$ $ \iff$ $A$ is invertible and $B$ is invertible. ■