Matrix Cancellation Property: Conditions for Equality of Matrices A and B

[matematikawan]

Let A,B be mxn matrices and C be nxk matrix. What is the necessary or sufficient condition such that AC=BC implies A=B ?

In my work, A and B are m by m matrices and C is just a column vector m by 1. In this specialized case, what are the condition imposed on the elements of C such that AC=BC will implies A=B.

Any clue please?

 

[tiny-tim]

hi matematikawan

you could start by simplifying AC=BC

 

[matematikawan]

I'm still not clear what to do.
Let say, I change C to X=(x₁ ... x_m)^t not a zero vector.


AX = BX so CX = 0 where C=A-B = (c_ij).
I want C to be zero m by m matrix so that A=B.

I have these equations.
c_{11}x_1 +c_{12}x_2 + ... +c_{1m}x_m = 0
c_{21}x_1 +c_{22}x_2 + ... +c_{2m}x_m = 0

etc.

How do I conclude that all c_ij are zero ?

 

[matematikawan]

Probably my question about cancellation doesn't make sense. I will be satisfied also if I able to express A in term of B and C.

AC = BC

What is A in term of B and C? Remember C is a column vector. So there is no inverse for C.

 

[tiny-tim]

hi matematikawan!

(just got up :zzz: …)

no, C isn't a column vector, it's a quite general matrix …

Let A,B be mxn matrices and C be nxk matrix. What is the necessary or sufficient condition such that AC=BC implies A=B ?

my suggestion is that you rearrange "AC=BC implies A=B"

to "(A-B)C = 0 implies A-B = 0" …

which, since A and B are completely general, is the same as "AC = 0 implies A = 0"

under what conditions (on m n k and C) will AC only be 0 when A is 0?

 

[matematikawan]

Thanks tiny-tim for the respond.

AC=0
A=0 if C has an inverse. That means C must be a square matrix m by m.

But my problem is that C is a column vector. The size of A, B and C already fix.
Actually I don't know why my question has been moved to the homework forum. I create the question myself. Probably it looks like an exercise in linear algebra.

Actually I'm trying a numerical method to solve pde using 'operational matrix of integration'. While plugging one of the boundary value, I need to solve for A in this equation.
AC=BC
A = ?

Thats why I initially posted the problem at linear algebra forum. Hoping for the answer. Normally I post question at DE forum.

 

[tiny-tim]

hi matematikawan!

start with C a non-zero column matrix …

obviously, there are plenty of non-zero A such that AC = 0, so that C doesn't work

now what about C a non-zero two-column matrix …

apart from a square matrix, again that C doesn't work

but as you say it can work for C a square matrix …

so try considering the tall-thin and short-fat categories separately

… Actually I don't know why my question has been moved to the homework forum

i expect it was moved because of the general pf policy that anything that could be homework goes in the homework forums

 

[matematikawan]

so try considering the tall-thin and short-fat categories separately

... This will take sometime for me to figure out this.


I search the internet and found out something call pseudoinverse.
A*C=B*C
So that A=B*C*pinv(C)

except that the answer is not unique. (to be expected since we have more unknowns than the equations)