I have this confusing question. I do not only need guidance in this question but I need a specific pattern which can help me solve these types of questions. If anyone would provide that, I would be grateful.
The Augmented Form in this Question is: $$ \left[ \begin{array}{cc|c} 1&-3&h\\ -2&6&-5 \end{array} \right] $$
MY TRIES:
First Approach:
- $R_2\to2R_1+R_2$
- $-5+2h=0\implies h=\frac{5}{2}$
- Therefore, the linear system is consistent only when $h$ is $\frac{5}{2}$.
Second Approach:
If we look closely, the first two elements of the second row are $-2$ times the first two elements of the first row, therefore, the last element of the second row must be $-2$ times $h$ (that is the last element of the first row). This, can't be true because the last element of the second row is $-5$ which is not the multiple of $2$. Therefore, the linear system is inconsistent for all the values of $h$.
Now, I am confused that which approach should I follow and also I am not sure that even if these approaches are true or is there any other way? Any help would be really appreciated. Thanks.