For what value K
is the following system consistent?
Source: Terry Stickel, Challenging Math Problems, 2015
For the system to be consistent it must be overdetermined, which implies the vectors representing the equations; (1,1,6), (k,1,18), (1,k,30); must be linearly dependent. Which then implies the determinant formed from the three vectors must equal zero.
| 1 1 6 |
| k 1 18 |
| 1 k 30 |
The determinant evaluates to 6k^2-48k+42=0. This equation has two roots k=1 and k=7. Checking each root shows that only k=7 actually makes the system consistent.