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.
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