Problem 4 [3 points]. A linear system $Ax = b$ is considered in this problem, where
$A = \begin{bmatrix} 2 & 1 & 1 \\ 4 & 1 & 1 \\ 6 & 2 & 1 \end{bmatrix}$, $b = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ and $x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$
4.1 Obtain all the nine cofactors of A.
4.2 Show that det A = 2 by utilizing the cofactors from Prob. 4.1.
4.3 Obtain the inverse of A by utilizing the cofactors from Prob. 4.1, and show that $AA^{-1} = I$.
4.4 Calculate the solution by $x = A^{-1}b$ to show that $x = -1.5, y = 7, z = -2$.
4.5 Verify the solution obtained in Prob. 4.4 by Cramer's rule.
4.6 Obtain $A^{-1}$ by means of the Gauss-Jordan elimination technique with normalization. For full credits, specify the row operation details in each step during the elimination process.
4.7 Verify that det A = 2 by Gauss elimination method or by referring to those in Prob. 4.6.
