Problem, illustrate ways in which the algebra of matrices is not analogous to the algebra...

Problem, illustrate ways in which the algebra of matrices is not analogous to the algebra of real numbers.

Find a 2 × 2 matrix A with each element +1 or −1 such that A2 = 0. The formula of Problem 1 may be helpful.

Problem 1

If , then show that

where I denotes the 2 × 2 identity matrix. Thus every 2 × 2 matrix A satisfies the equation

A2 − (trace A) A + (det A) I = 0

where det A = ad − bc denotes the determinant of the matrix A, and trace A denotes the sum of its diagonal elements. This result is the 2-dimensional case of the Cayley Hamilton theorem of Section 6.3.