# Where to begin when comparing matrices and their inverses

I've just started with linear algebra and am NOT looking for the answer. I'm just looking for a way to begin answering the following true/false question:

If A,B,C are matrices of the same size such that A+C=B+C then A=B and A(−1) =B(−1)

The A(-1) and B(-1) are the inverses of A and B.

Should I create, let's say, a 2x2 matrix (with any numbers) for A, B, and C?

If so, then what would be my next step?

## 1 Answer

Looking at e.g. the $2\times 2$ case may give an intuition whether a statement is wrong or false and might even give an idea how to prove/disprove the statement rigorously. Nevertheless, this does not give a proof.

How would you proceed in case that $A,B,C$ would not be matrices but, let us say real numbers?

And moreover, why can we even say what the inverse of the matrix $A$ is?

• I'm not sure I'm understanding your comment. Why would I want the matrices to be real numbers? – Phatfoo Aug 4 '18 at 14:15
• Real numbers are just one-dimensional matrices. Also the set of matrices of fixed size as well as the real numbers form a group with respect to addition. The first part of the statement can be generalized to arbitrary groups. I was proposing to look at the real numbers because the reals are a maybe easier to understand or to have intuition whether a statement holds. Moreover, I assume that you are working with matrices over some field?! – Jonas Lenz Aug 4 '18 at 14:22