Let A be a nonzero 2 x 2 matrix and write U X in M22 XA A

Let A be a nonzero 2 x 2 matrix, and write U = {X in M22 | XA = AX} Show that dim U >= 2

Solution

One apparent solution for the matrix equation XA = AX is X = I₂ regardless of the entries in A.

Further, if A is a non-zero 2x2 matrix, there are two possibilities; either det(A) 0 and hence A is invertible or det A = 0. If det A 0, then, since XA = AX, on right multiplication by A^-1, we have XAA^-1   = AXA^-1 , or X = AXA^-1 .

In case det (A) = 0, our approach can be as under: Let X have [ p, q] and [ r, s] as its 1^st and 2^nd rows. Since A is a non-zero 2 x 2 matrix, at least one entry in A is non-zero. We shall consider 4 cases as under:

In view of the above, dim U 2