Matrices can behave quite differently from real numbers.

Question

Matrices can behave quite differently from real numbers. It is possible that for square matrices A and B, the matrix product AB is not equal to the product BA. When they do happen to be equal, we say that A and B commute. In other words, A and B commute if AB = BA.

(a) Find all 2 × 2 matrices A that commute with B = | 0 1

1 0 |.

(b) Now suppose that A and B are square matrices of any size. Prove that AB = BA if and only if (A − B)(A + B) = A2 − B2

Answer 1

(a). Let A be the 2 x 2 matrix with first row a b and second row c d.

Then, AB is the 2 x 2 matrix with first row b a and second row d c.

BA is the 2 x 2 matrix with first row c d and second row a b.

For the equation AB = BA to be true, all corresponding entries of AB and BA must be equal.

So, we see that b = c and a = d.

Therefore, A must be a 2 x 2 matrix where the main diagonal entries are equal and the entries on

the other diagonal are also equal.

An example would be the 2 x 2 matrix with first row 5 -3 and second row -3 5.

(b). Suppose that A and B are both n x n matrices.

If AB = BA, then (A - B)(A + B) = A² + AB - BA - B² = A² - B².

If (A - B)(A + B) = A² - B², then A² + AB - BA - B² = A² - B^2.

Subtracting A² from both sides and adding B² to both sides gives us AB - BA = n x n zero matrix.

So, adding BA to both sides yields AB = BA.

Therefore, AB = BA iff (A - B)(A + B) = A² - B².