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--- lecture_14 [2017/03/07 15:09] – rupert
+++ lecture_14 [2017/03/09 10:49] (current) – [Corollary on invertibility] rupert
@@ Line 90: / 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. ■