2x2 matrix help please!

let A be [[1 2] and B be [[-3 2]
............[2 4]]................[-1 -1]]
Find a matrix C for which the (2,2) entry is 0 and such that AB=AC and BA=CA

Let C=[a b]
..........[c 0]


AB=[-5 0]
.......[-10 0]

AC=[a+2c b]
......[2a+4c 2b]


Since AB=AC then

Then a+2c=-5 and 2a+4c=-10 which is the same equation.b=0

BA=[1 2]
.......[-3 -6]

CA=[a+2b 2a+4b]
.......[c 2c]

You know that b=0 and since BA=CA then

a+2b=1 and 2a+4b=2 which are the same equation, then a=1

c=-3

Then C=[1 0]
..............[-3 0]

C matrix will look somewhat like this:
[X Y]
[Z 0]

You must find X,Y,Z and the conditions are specified.

AB=[-5 0]
[-10 0]

AC= [X+2Y Y ]
[2X+4Z 2Y]

Since AB=AC,
Y=0
X=-5
Z=0.

Substitute the values in matrix C=[X Y]
[Z 0]