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--- lecture_14 [2017/03/09 10:44] – [Theorem: row/column operations and determinants] 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 $\det(AB)=\det(A)\det(B)$
+==== Corollary on invertibility ====
+  - $A^T$ is invertible if and only if $A$ is invertible
+  - $AB$ is invertible if and only if **both** $A$ and $B$ are invertible
+=== Proof ===
+  - 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. ■