Solving AB=0 for B given A Matrix

[Dell]

gven a matrix A=

1 0 2
1 1 1
5 2 8

and knowing AB=0 ,B\neq0
what are possible values of B

is there any way to solve this other than
making B a matrix of parameters , doing the multiplication and solving, ie
B=

x y z
a b c
d e f

x+0a+2d=0
y+0b+2e=o
... etc

 

[Mark44]

It's helpful to know about the nullspace of a matrix in this problem. In general, the nullspace is the set of vectors x such that Ax = 0.

For this problem, the nullspace is one-dimensional, and consists of all scalar multiples of (-2, 1, 1).

Instead of looking at AB = 0, think about what's happening to the individual columns of B, call them B_1, B_2, and B_3. What can you say about A*B_1 = 0? A*B_2 = 0? A*B_3 = 0?

 

[Dell]

all got to be multiples of (-2 1 1)? am i on the right track, haven't yet learned about nullspace.

 

[Mark44]

Yes and yes, so congratulations! Keep in mind that the columns are different multiples of (-2, 1, 1). (Hint: use parameters.)

To check, write a matrix B as above and calculate AB. Should come out with the 3 x 3 zero matrix.