Question $9$: $3$ Marks

Evaluate the determinant given below $by$ reducing $itto$ row echelon form:

$|[1,0,2,1],[0,1,1,1],[2,0,1,0],[0,1,2,3]|$

Question $10$: $4$ Marks

Given that

$|[a,b,c],[d,e,f],[g,h,i]|=-6$

Evaluate det$(A)$ and det$(B)$ for

$A=\left[\begin{array}{ccc}-a& -b& -c\\ 2d& 2e& 2f\\ -3g& -3h& -3i\end{array}\right]$ and $B=\left[\begin{array}{ccc}3g& 3h& 3i\\ 2a+d& 2b+e& 2c+f\\ d& e& f\end{array}\right].$

Question $11$: $4$ Marks

Given a matrix

$A=\left[\begin{array}{ccc}3& 1& -2\\ 2& 1& -1\\ -1& 0& -2\end{array}\right]$

Use the reduced row echelon form $to$ determine

$(a)$ det$(-\frac{2}{3}A)$

$(b)$ det$(-\frac{2}{3}{A}^T)$

$(c)$ det$(-3A^{-1})$

$(d)$ det$(A^4).$

Question $12$: $5$ Marks

Use Cramer's rule $to$ solve for $uin$ the system

$2u+x+y+z=3$

$-8u-7x-3y+5z=-3$

$u+4x+y+z=6$

$u+3x+7y-z=1$

Use row reduction $to$ evaluate the determinants before solving for $u,$ without solving for $x,y$ and $z.$