The augmented matrix is given as

\[\begin{bmatrix}1&h&5\\-2&6&-15\end{bmatrix}\]

Reduce this augmented matrix to echelon form

Consider

\[\begin{bmatrix}1&h&5\\-2&6&-15\end{bmatrix}\]

\(\displaystyle{R}_{{2}}\to{R}_{{2}}+{2}{R}_{{1}}\)

\[\begin{bmatrix}1&h&5\\0&6+2h&-5\end{bmatrix}\]

This system is inconsistent if and only if

\(\displaystyle{6}+{2}{h}={0}\)

\(\displaystyle{2}{h}=-{6}\)

\(\displaystyle{h}=-{3}\)

Hence this system is consistent if and only if \(\displaystyle{h}\ne-{3}\)

Therefore option (A) is correct.

\[\begin{bmatrix}1&h&5\\-2&6&-15\end{bmatrix}\]

Reduce this augmented matrix to echelon form

Consider

\[\begin{bmatrix}1&h&5\\-2&6&-15\end{bmatrix}\]

\(\displaystyle{R}_{{2}}\to{R}_{{2}}+{2}{R}_{{1}}\)

\[\begin{bmatrix}1&h&5\\0&6+2h&-5\end{bmatrix}\]

This system is inconsistent if and only if

\(\displaystyle{6}+{2}{h}={0}\)

\(\displaystyle{2}{h}=-{6}\)

\(\displaystyle{h}=-{3}\)

Hence this system is consistent if and only if \(\displaystyle{h}\ne-{3}\)

Therefore option (A) is correct.