Prove or disprove: A^2 - B^2 = (A - B)(A + B) for all square matrices A and B of the same size.

Question

Prove or disprove: A² - B² = (A - B)(A + B) for all square matrices A and B of the same size.

Answer 1

The statement is FALSE. The difference of squares pattern

DOES NOT apply to Matrices and it DOES NOT WORK with matrices.

Here is one counter-example of the infinitely many!!

A B

1 0 2 3

1 2 3 2

A^2 = 1 0 1 0

1 2 1 2

= 1 0

3 4

B^2 = 2 3 2 3

3 2 3 2

= 13 12

12 13

A^2-B ^2 = -12 -12

-9 -9

A+B = 3 3

4 4

A-B = -1 -3

-2 0

(A+B)(A-b) = 3 3 -1 -3

4 4 -2 0

= -12 -9

-17 -12

FAILS!!!

THe problem is that matrix multiplication is

NOT COMMUTATIVE..

The difference of squares pattern says:

(A+B)(A-B) = A*A - A*B + B*A - B*B

= A^2 - A*B + B*A - B^2

However, commutative property is needed

to switch the order of the matrix multiplication

which is invalid