If detA detB Show that it is not necessarily true that A is

If det(A) = det(B), Show that it is not necessarily true that A is similar to B

Solution

If det(A) = det(B), Show that it is not necessarily true that A is similar to B. To prove this, a counter example will be enough. We know that two matrices A and B are similar if and only if there exists an invertible matrix U such that UAU^-1 = B. By this definition, we can be certain that I₂ and I₃ are not similar. However, det (I₂ ) = det ( I₃ ) = 1.