(a) Perform the LU-factorization with column pivoting of the matrix

$$A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ 4 & 1 & 0 \\ -2 & 2 & 1 \end{array}\right]$$

(b) Explain briefly why every nonsingular matrix $A$ admits an LU-factorization with column pivoting.