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# FP2 Matrices Help

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1. For part i) determinant is 1/(5-k)

in part 3) k = 5 which means determinant is 0, how is it possible for it to have solutions? How do I find them?
2. The fact that the determinant is zero means that there is not a unique solution, not that there are no solutions. There might not be, but there again, there might be an infinite number of them. All that you know is that there is not one solution.

Choose two of the equations, and eliminate one of x, y or z (for example, take two lots of the first equation away from the second to eliminate y). Now take the other equation along with one of the ones that you have already used and eliminate the same variable again (for example, take five lots of the first equation away from the third to again eliminate y). If you have done this correctly, the left hand sides of the equations you have obtained will be the same. The only way that the equations can have any solutions (in which case they will have an infinite number of solutions) is if the right hand sides are also equal. This will lead you to a simple linear equation in p.

There are other ways to do this, but I personally prefer sticking with the equations rather than doing tricks with matrices - I feel more involved with the actual problem this way!
3. There is no unique solution as there is no inverse to the coefficient matrix. However you can still find the general solution- where all the solutions lie on a single line.
There are many ways of doing this. Normal row operations, Jordan Gauss the most common ones I would use.

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