1. ## matrix multiplication commutativity

Let A & B be matrices such that:

A.B = B.A (with det(B) =/= 0)

Show that B can be written under the form:

B = mA + pI where m & p are real numbers and I is the identity matrix.

2. Originally Posted by tombrownington
Let A & B be matrices such that:

A.B = B.A (with det(B) =/= 0)

Show that B can be written under the form:

B = mA + pI where m & p are real numbers and I is the identity matrix.
The two matrices that A always commutes with are itself and I.

If A commutes with B it therefore follows that B must be a linear combination of A and I.

3. Sorry Mr. Fantastic, I didn't quite get you.
Can you elaborate a little.

4. Originally Posted by tombrownington
Sorry Mr. Fantastic, I didn't quite get you.
Can you elaborate a little.
Please be specific - what don't you get?

5. Originally Posted by mr fantastic
The two matrices that A always commutes with are itself and I.

If A commutes with B it therefore follows that B must be a linear combination of A and I.
I don't follow the "it therefore follows that B must be a linear combination of A and I" part. That was the whole point of the question in any case.

6. Originally Posted by tombrownington
I don't follow the "it therefore follows that B must be a linear combination of A and I" part. That was the whole point of the question in any case.
You need to make the observation that A commutes only with itself (that is, with A) and I. Once this has been noted, it clearly follows that if A commutes with B then B must be a linear combination of the matrices that commute with A, that is, a linear combination of the matrices A and I.

7. Originally Posted by tombrownington
Let A & B be matrices such that:

A.B = B.A (with det(B) =/= 0)

Show that B can be written under the form:

B = mA + pI where m & p are real numbers and I is the identity matrix.
This result is not true. For example, suppose that A is the identity matrix and B is any invertible matrix that is not a scalar multiple of the identity.

8. Originally Posted by Opalg
This result is not true. For example, suppose that A is the identity matrix and B is any invertible matrix that is not a scalar multiple of the identity.
Good point!