Consider the system of linear equations
$$\mathbf A \mathbf x = \mathbf b,$$
where $\mathbf A \in \mathbb R^{n \times n}$ is invertible and $\mathbf b \in \mathbb R^n$. Let $\mathbf D \in \mathbb R^{n \times n}$ be a diagonal and invertible.
We can solve the system $\mathbf A \mathbf x = \mathbf b$ directly. However, I want to show that if we solve it such a way that we calculate $\mathbf D \mathbf A \mathbf D$ and $\mathbf D\mathbf b$, solve the system $\mathbf D \mathbf A \mathbf D \mathbf y = \mathbf D\mathbf b$, and calculate $\mathbf x = \mathbf D \mathbf y$. Then this $\mathbf x$ will be the same $\mathbf x $ as solving it directly.
Using Algorithm GEPP (Gaussian elimination with partial pivoting), how would you go about showing that the vector $\mathbf x$ calculated in Approach 2 indeed solves $\mathbf A \mathbf x = \mathbf b$.
Thank you in advance.
if we solve your second equation, we get :
$$y=\left(DAD \right)^{-1}Db=D^{-1}A^{-1}D^{-1}Db=D^{-1}A^{-1}b $$
so you get :
$$Dy=A^{-1}b=x $$ by your first definition