As I said in my previous question I'm studying Linear Algebra. I don't know the answer to the following question and want help.
True or false. The matrix [[a,b],[c,-a]] with determinant 1 and c not 0 is similar to [[0,-1],[1,0]] up to sign.
What I tried: since it is a 2x2 matrix I multiplied by [[x,y],[z,w]] on left of the zero-one matrix by the inverse on the right. But I couldn't conclude.
Is there a better approach? Is it true or false?
Thanks in advance.
$A=\pmatrix{a & b \\ c & -a}$ is similar to $B=\pmatrix{0 & -1 \\ 1 & 0}$, if and only if there is an invertible matrix $S=\pmatrix{s_1 & s_2 \\ s_3 & s_4}$ with $AS=SB$. So compute both $AS$ and $BS$. This gives $4$ linear equations in the variables $s_1,\ldots ,s_4$. Solve these equations, using $a^2+bc=-1$, which says that $\det(A)=1$. Then you will find solutions for $S$.