Decomposing the “sign flip” matrix in terms of Pauli matrices

Question

Can the matrix:

$A=\left[\begin{matrix}0 & -1 \\ -1 & 0\end{matrix}\right]$

be somehow expressed as a product of the $3$ standard Pauli matrices?

I'm being able to diagonalize $A$, but not sure if it can be expressed in terms of Pauli-X, Y, Z.

Answer 1

$\begin{bmatrix} 0 & -1\\ -1 & 0\end{bmatrix} = \begin{bmatrix} 0 & 1\\ 1 & 0\end{bmatrix} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix} \begin{bmatrix} 0 & -i\\ i & 0\end{bmatrix} \begin{bmatrix} 1 & 0\\ 0 & -1\end{bmatrix} $