Problem $1$: Given the system of equations

$10x_1+2x_2-x_3=27$

$-3x_1-6x_2+2x_3=-61\mathrm{.}5$

$x_1+x_2+5x_3=-21\mathrm{.}5$

Solve by Gauss elimination with partial pivoting. Calculate the determinant as part of the computation. Show hand calculations of all steps of the computation.

Hints:

Choose the largest magnitude coefficient as the pivot.

Find the maximum magnitude element in each column.

Swap rows if needed to ensure the pivot is the largest.

Calculate the determinant.

Proceed with elimination and equation updates.