Here is Prob. 8, Sec. 2.1, in the book _Linear Algebra With Applications by Steven J. Leon and Lisette de Pillis, tenth edition:
Show that if $E$ is an elementary matrix, then $E^T$ is an elementary matrix of the same type as $E$.
Here is an answer to this question. However, I am not sure how to rigorously verify the three bulleted statements in the answer.
My Attempt:
Let $E$ be an $n \times n$ elementary matrix. Then there are the following three cases:
Let us denote the rows of $I_n$ by $\overline{\mathbf{e}}_i$ for $i = 1, \ldots, n$. Then we have $$ \overline{\mathbf{e}}_i = \left[ \begin{matrix} 0 & \ldots & \underbrace{1}_{\mbox{$i$th entry}} & \ldots & 0 \end{matrix} \right]. $$ Let $\overline{\mathbf{r}}_i$, where $i = 1, \ldots, n$, denote the rows of $E$.
Case 1. $E$ is obtained from the $n \times n$ identity matrix $I_n$ by interchanging rows $i_0$ and $j_0$, where $i_0$ and $j_0$ are some fixed integers such that $1 \leq i_0 < j_0 \leq n$. Then, for each $i = 1, \ldots, n$, we have $$ \overline{\mathbf{r}}_{i} = \begin{cases} \overline{\mathbf{e}}_{j_0} & \mbox{ if } i = i_0, \\ \overline{\mathbf{e}}_{i_0} & \mbox{ if } i = j_0, \\ \overline{\mathbf{e}}_{i} & \mbox{ if } i \neq i_0 \mbox{ and } i \neq j_0. \end{cases} $$
Am I right? If so, then how to proceed from here?
Case 2. If $E$ is obtained by multiplying row $i_0$, where $i_0$ is some fixed integer such that $1 \leq i_0 \leq n$, of $I_n$ by some non-zero scalar $\alpha$, then, for each $i = 1 \ldots, n$, we have $$ \overline{\mathbf{r}}_i = \begin{cases} \alpha \overline{\mathbf{e}}_{i_0} & \mbox{ if } i = i_0, \\ \overline{\mathbf{e}}_{i} & \mbox{ if } i \neq i_0. \end{cases} $$
How to proceed from here?
Case 3. $E$ is obtained from $I_n$ by adding a scalar $\alpha$ times row $j_0$ of $I_n$ to row $i_0$, where $i_0$ and $j_0$ are some fixed positive integers not exceeding $n$ such that $i_0 \neq j_0$. Then, for each $i = 1, \ldots, n$, we have $$ \overline{\mathbf{r}}_i = \begin{cases} \overline{\mathbf{e}}_{i_0} + \alpha \overline{\mathbf{e}}_{j_0} & \mbox{ if } i = i_0, \\ \overline{\mathbf{e}}_{i} & \mbox{ if } i \neq i_0. \end{cases} $$
How to proceed from here?