Matrices

[math magic]

Could someone please help with the following questions. I’m having some trouble with them. It would be greatly appreciated. Thanks.

Matrices P and Q are both 2x2 matrices and P=Q+PQ+PQ2.
Determine P given that Q=
-1 0
5 -2

x + 3y – z = 5
2x + 7y – 3z = 6
-x + 4y + (p2 – p – 6)z = p – 33
For what values of p will the above system of equations have:
a) an infinite number of solutions
b) no solutions
c) a unique solution? Find this solution in terms of p.


The scores obtained by Adam, Bob, Cheng and Diana in a maths test were given in the following form.
The average of the four marks was 68.5. The sum of the marks of Bob and Cheng was 2 more than the sum of the marks of Adam and Diana. Bob scored higher than Adam with the difference of their two marks equal to one eight of Diana’s marks. Finally the sum of the marks of Adam and Cheng plus twice the sum of the marks of Bob and Diana was 423.
Construct the system of linear equations for the above information and represent it in matrix form.