# Find two 2x2 matrices that multiply to give 0

Physics Slayer
Homework Statement:
Find two ##2x2## matrices ##A## and ##B## such that ##AB = 0## but ##BA\neq0##
Relevant Equations:
AB=0
One way would be to assume
$$A= \begin{bmatrix}a_1 & a_2\\a_3 & a_4 \end{bmatrix}$$ and $$B=\begin{bmatrix}b_1 & b_2\\b_3 & b_4\end{bmatrix}$$ and then multiply but then you end up with 4 equations and 8 variables, how would that work?

the other way would be to use trial and error, any help would be appreciated.

Staff Emeritus
Gold Member
I would think of it in terms of images and kernels.

To start, what do the rank of A and B need to be?

topsquark
Physics Slayer
I would think of it in terms of images and kernels.

To start, what do the rank of A and B need to be?
I am unfamiliar with terms like images and kernels.

both A and B are 2x2 matrices

Staff Emeritus
Gold Member
Do you know what the rank of a matrix is?

topsquark
Physics Slayer
Do you know what the rank of a matrix is?
I thought its 2x2 its given in the question

Homework Helper
do you know that AB = 0 means the rows of A are perpendicular to the columns of B?

Maarten Havinga, WWGD and topsquark