The formula in Problem 1 can be used to compute A2 without an explicit matrix multiplicati...

The formula in Problem 1 can be used to compute A2 without an explicit matrix multiplication. It follows that

without an explicit matrix multiplication,

,

and so on. Use this method to compute A2, A3, A4, and A5 given

.

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.