Finding the expression of the inverse of $(AB)^T$

Question

I know that $(AB)^T$ = $B^TA^T$ and that $(A^T)^{-1}= (A^{-1})^T$ but couldn't reach any convincing answer. Can someone demonstrate the expression.

Answer 1

you have the next property too $$(AB)^{-1}=B^{-1}A^{-1}$$ then the inverse of $(AB)^{T}$ is $\left[(AB)^{T}\right]^{-1}$ $$\left[(AB)^{T}\right]^{-1}=\left[B^{T}A^{T}\right]^{-1}=\left[A^{T}\right]^{-1}\left[B^{T}\right]^{-1}=\left[A^{-1}\right]^{T}\left[B^{-1}\right]^{T}$$

Answer 2

One way to see, you can compare entries of two matrices $(AB)^t$ and $B^tA^t$.

Firstly, we have $$AB_{ij} = \sum_{k} a_{ik}b_{kj}.$$

When to transpose a matrix, we switch the rows and the columns. $$(AB)^t_{ij} = AB_{ji} = \sum_{k} a_{jk}b_{ki}.$$

Now, let see the entry ij of $B^tA^t$:

$$B^tA^t_{ij} = \sum_{k} B^t_{ik}A^t_{kj} = \sum_{k} b_{ki}a_{jk}.$$

Finally, you can apply this to see $(A^t)^{-1} = (A^{-1})^t$.

$$I = I^t = (AA^{-1})^t = (A^{-1})^tA^t.$$