Answer to Question #255443 in Abstract Algebra for Tami

Question

The subtraction of a matrix B may be considered as the addition of the marix (-1)B.Does the cummutative law of addition permit us to state that A-B=B-A?If not,how would you correct the statement?

Answer 1

The commutative law of addition does not permit us to state that A-B = B-A and here is why.

As we can see:

• A-B = A+(-1)B = (-1)B+A
• B-A = B+(-1)A = (-1)A+B
• (-1)B+A != (-1)A+B

To correct the statement "A-B = B-A", let's try transforming it's left part:

A-B = A+(-1)B = (-1)((-1)A+B) = (-1)(B-A) = -(B-A)

Thus, "A-B = -(B-A)" is the corrected statement.