[jhomie]

Hi,

I'm doing some maths revision over the summer and once chapter is matrices, but there are a few questions I don't know how to do.

The book shows the answer, but I don't know how to get it.

Q.11

Q.12 Sorry for it saying x,y and ztwice

Q.14

Q.15

Q.17


If you can show me how to do these that would help me understand them a lot

[generalebriety]

11. Well, the matrix is singular (non-invertible), so we know there's either no solutions at all, or infinitely many. If there's infinitely many, the three equations represented by that one matrix equation are consistent, but linearly dependent. Since this question's proving tricky for you, and there's a lot like it, I'll do this one for you:

Take the third equation, (3) 2x + y - 3z = k. We're gonna try to use the other two equations in a sort of 'simultaneous equations' way to reduce the left hand side to zero. The two equations are:
(1) 2x - y - 9z = 7
(2) x + 2y + 3z = 1.
Subtract (1) from (3) (because I want to get rid of the x term) and get
(4) 2y + 6z = k - 7.
Subtract 2*(2) from (1) (because I want to get rid of the y term in (4), so I'm getting rid of the x term in something else to help) and get
(5) -5y - 15z = 5
Multiply (4) by 5 and (5) by -2:
(4) 10y + 30z = 5k - 35
(5) 10y + 30z = -10
and we can now subtract one from the other (or compare their right hand sides).

12. Again, you're meant to rewrite as a matrix equation and show the matrix is singular in part a. It's then back to Q11 for part b. Part c should be straightforward (and I'm reluctant to say how I'd do it, because it's the sort of thing that gets taught in a million different ways).

I've looked at the other three and, as long as you've understood my explanations above, you should be able to do them yourself now with a bit of work.