Answer
See solution
Work Step by Step
Write augmented matrix for given system and row decreased it:
$\left[\begin{array}{ll|l}1 & 3 & k \\ 4 & h & 8\end{array}\right],$ multiply first row with -4 and add to second $\sim\left[\begin{array}{cc|c}1 & 3 & k \\ 0 & h-12 & 8-4 k\end{array}\right]$
Solution is unique if second column is pivot, infinite number of solutions if there is no pivot in second row and no solution if third column is pivot.
Unique:
$h-12 \neq 0$
$h \neq 12$
Infinite number:
$h-12=0$
$h=12$
$8-4 k=0$
$k=2$
No solution:
$h-12=0$
$h=12$
$8-4 k \neq 0$
$k \neq 2$
a)
Write augmented matrix for given system and row reduce it:
$\left[\begin{array}{cc|c}-2 & h & 1 \\ 6 & k & -2\end{array}\right],$ multiply first row with 3 and add to second $\sim\left[\begin{array}{cc|c}-2 & h & 1 \\ 0 & k+3 h & 1\end{array}\right]$
Solution is unique if second column is pivot, infinite number of solutions if there is no pivot in second row and no solution if third column is pivot.
Unique:
$k+3 h \neq 0$
$k \neq-3 h$
Infinite number:
Not possible because element in second row, third column is 1 and
No solution:
$k+3 h=0$
$k=-3 h$
