Having Fun with Eigenvalues

If A is a 3×3 matrix with eigenvalues 2, 3, 5, and B is its inverse matrix. Then:

Trace of A is 2 + 3 + 5 = 10.

Determinant of A is (2)(3)(5) = 30.

Trace of B is 1/2 + 1/3 + 1/5 = 1.033…

Determinant of B is (1/2)(1/3)(1/5) = 1/30 = 0.033…

Clean and sweet!

Theorem: If A is an n × n matrix, then the sum of the n eigenvalues of A is the trace of A and the product of the n eigenvalues is the determinant of A.

If B is the inverse matrix of A, the sum and product of the inverse eigenvalues of A are, respectively, the trace and determinant of B.

Reference

Facts About Eigenvalues, by David Butler