Show that if A is a 2 × 2 matrix such that AB = BA whenever B is a 2 × 2 matrix, then A = cI. where c is a real number and I is the 2 × 2 identity matrix.
SOLUTIONStep 1In this problem, we are asked to show that the matrix whenever AB=BA.Step 2We know that if AB=BA then A and B are simultaneously diagonalizable.This implies that there exist a nonsingular matrix P such that are diagonal matrices.Therefore is a diagonal matrix.Step 3Now we have to show that in all the diagonal elements are same.To show this let us use matrix S[i,j] such that and and all other .Therefore...
Textbook: Discrete Mathematics and Its Applications
Author: Kenneth Rosen
This full solution covers the following key subjects: Matrix, show, real, Identity, such. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. Since the solution to 39E from 2.SE chapter was answered, more than 239 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 39E from chapter: 2.SE was answered by , our top Math solution expert on 06/21/17, 07:45AM. The answer to “Show that if A is a 2 × 2 matrix such that AB = BA whenever B is a 2 × 2 matrix, then A = cI. where c is a real number and I is the 2 × 2 identity matrix.” is broken down into a number of easy to follow steps, and 42 words. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.