1)

x - y + 2z = 0

3x + 4y + az = 0

2x + by + 4z = 0

show that the system of equations has a non-trivialsolution either a = 6 or b = -2.

Find the general solution to the system of equations whena = 6.

2)

Let A =

7 -8 0

4 -5 0

-1 1 1

If possible, find a diagonal matrix C and an invertiblematrix B such that A = BCB^-1

I have found the characteristic polynomial of A, alongwith the eigenvalues/vectors. However I just don't understand thisquestion, and my notes/textbook is completely useless! I assume thetwo matrices A and C will be similar, thus have the sameeigenvalues. But how am I supposed to show that matrix!?

3)

Consider you have a 3x3 matrix like this:

a b 0

0 a b

0 0 a

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