consider two n x n matrices A and B with real entries Is it

consider two n x n matrices, A and B, with real entries. Is it necessarily true that A² - B² = (A - B)(A + B)? justify your answer.

Solution

A² - B²

Lets consider

A = 1    2

       3   4

B = 2    3

       5    7

Now A² = 7   10

               15   22

B²   = 19    27

           45    64

A² -B² = -12   -17

               -30   -42

A - B = -1 -1

            -2 -3

A+B = 3    5

           8   11

(A - B)(A + B) =   -11   -16

                            -30   -43

so it is not true as they are not equal