Thread: 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.

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.

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

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?

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.

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.

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.

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!