Problem 1

(a). Let and Verify that and Have the Same Eigenvalues

and have the same characteristic polynomial meaning they must also have the same eigenvalues.

(b). Prove That, for Any Matrices and where is Invertible, and Have the Same Eigenvalues

We can see that both and must have the same eigenvalues.

Problem 2

A left stochastic matrix is a square matrix with nonnegative entries whose entries in each column sum to 1.

(a). Find the Eigenvalues of the Left Stochastic Matrix

(b). Let Verify that and Have the Same Eigenvalues by Finding Those Eigenvalues

The determinant of a matrix and it's transpose are equal so we can conclude that they have the same eigenvalues.

(c). Let Be an Matrix. Prove that and Have the Same Characteristic Polynomial

The characteristic polynomial is defined as for a matrix

The determinant of a matrix and it's transpose are equal so we can conclude that they have the same eigenvalues.

(d). Use part (c) to Prove that 1 is an Eigenvalue of Every Left Stochastic Matrix

Since all the columns in a left stochastic matrix sum to 1 taking the transpose gives us a matrix where all the rows sum to 1

Which gives us a