Verify that the determinants of the following two matrices are equal to each other using only elementary row operations and without expanding the determinants.
\begin{bmatrix}a-b&1&a\\b-c&1&b\\c-a&1&c\end{bmatrix} \begin{bmatrix}a&1&b\\b&1&c\\c&1&a\end{bmatrix}
I attempted to solve this problem by using row swaps for both matrices such that the last column of each matrix is equal to each other. After that, I'm stuck and not exactly sure of how to continue.
$$\begin{bmatrix}a-b&1&a\\b-c&1&b\\c-a&1&c\end{bmatrix} $$
$R_1\rightarrow R_1 + (a+b+c)R_2$
$$\begin{bmatrix}2a+c&1&a\\2b+c&1&b\\2c+a&1&c\end{bmatrix}$$
$R_1\rightarrow R_1 -2R_3$
$$\begin{bmatrix}c&1&a\\a&1&b\\b&1&c\end{bmatrix}$$
And now changing 2 rows give desired result.