Question

Determine the​ value(s) of h such that the matrix is the augmented matrix of a consistent linear system

Answer 1

Given the matrix

$\left[\begin{array}{ccc}1&h&4\\3&6&8\end{array}\right]$

We perform the following matrix operations

$\left[\begin{array}{ccc}1&h&4\\3&6&8\end{array}\right] -3R_1+R_2\rightarrow R_2 \\ \\ \left[\begin{array}{ccc}1&h&4\\0&6-3h&-4\end{array}\right] \frac{1}{6-3h}R_2\rightarrow R_2 \\ \\ \left[\begin{array}{ccc}1&h&4\\0&1&\frac{-4}{6-3h}\end{array}\right] -hR_2+R_1\rightarrow R_1 \\ \\ \left[\begin{array}{ccc}1&0& \frac{24-8h}{6-3h} \\0&1&\frac{-4}{6-3h}\end{array}\right]$

Thus, the matrix is consistent for
$h\neq6-3h \\ \\ h+3h\neq6 \\ \\ 4h\neq6 \\ \\ h\neq \frac{6}{4} \\ \\ h\neq \frac{3}{2}$