Determine The Value Of H Such That The Matrix Is The Augmented Matrix Of A Consistent Linear System

The determinant of the coefficient matrix ( the 2 x2 matrix formed from the first two columns) is zero. This means that the two underling linear equations are not independent.
If h is chosen as -2, the two equations would be 6 x – 3 y = -2 and -12 x + 6 y = 4. The second of these is -2 times the first. ( Explicitly showing that they are not independent.) So, with h = -2, there are infinitely many solutions (x,y) all with y = 2 x + 2/3. Consistent but degenerate.

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For any other choice of h, the two equations have no solutions because they correspond to parallel lines. This would be considered an inconsistent linear system.
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