Find $b$ such that we know the solution $x$ of $Ax = b$

Question

So, $A$ is a $n$ x $n$ matrix where:

• All the diagonal elements are $1$
• All the elements on the lower triangular part are $-1$
• The elements on the $n^{th}$ column are $1$
• And every other element is $zero$

I am trying to find $b$ such that we already know what $x$ is without caring about the value of $n$.

I tried to use a $b$ similar to the one we use in $H_n x=b$ where $H_n$ is a $n$ x $n$ Hilbert matrix but still I couldn't specifically define the $x$.

Any hint would be appreciated!

Answer 1

I suggest that you try $b$ = first column of $A$; then $x = e_1$. Alternatively, pick $b$ = last column of $A$ and then $x = e_n$. Indeed, this works for any column. :)