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