$12\mathrm{.}1($a$)$ Solve the systems below by hand using Gaussian elimination and back substitution $($exactly as

above$)$ on the augmented matrix. As a by$-$product, give the LU decomposition of A$.$ Pivot wherever

appropriate $($the number being eliminated should be smaller than the number eliminating it$).$

$0\mathrm{.}5x_1+4x_2=0$

$2x_1+4x_2=-3$

Check by hand that $LU=PA$ and $Ax=b.$ If you do not permutate rows of matrix $A.P$ is the identity

matrix.

$($b$)$ Compare with Matlab. Turn in the MatLab output.

$12\mathrm{.}2($a$)$ Solve the systems below by hand using Gaussian elimination and back substitution $($exactly as

above$)$ on the augmented matrix. As a by$-$product, give the LU decomposition of A$.$ Pivot wherever

appropriate $($the number being eliminated should be smaller than the number eliminating it$).$

$x_1+2x_2+3x_3=2$

$4x_1+x_2+x_3=1$

$2x_1+3x_2+9x_3=1$

Check by hand that $LU=PA$ and $Ax=b.$ If you do not permutate rows of matrix $A.P$ is the identity

matrix.

$($b$)$ Compare with Matlab. Turn in the MatLab output.