Determine the values of h such that the matrix is

rompentskj 2021-11-23 Answered
Determine the values of h such that the matrix is the augmented matrix of a consistent linear system.
\[\begin{bmatrix}1&h&5\\-2&6&-15\end{bmatrix}\]

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Expert Answer

Witionsion
Answered 2021-11-24 Author has 1860 answers
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.
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