At least one of the answers above is NOT correct The matrix

At least one of the answers above is NOT correct. The matrix T has eigenvalues and eigenvectors: v_1 = [1 0 0], with lambda_1 = 1. V_2 = [2 1 1], with lambda_2 = 1/2. V_3 = [2 1 0], with lambda_3 = 2/3. Give formulas for the following: (A) T^n [2 1 1] = [2 1 1] (B) T^n [-6 -3 0] = [] (C) T^n ([3 0 0] + [6 3 3] + [8 4 0]) = (D) T^n [14 6 3] =

Solution

A·v=·v

In this equation A is an n-by-n matrix, v is a non-zero n-by-1 vector and is a scalar (which may be either real or complex). Any value of for which this equation has a solution is known as an eigenvalue of the matrix A. It is sometimes also called the characteristic value. The vector, v, which corresponds to this value is called an eigenvector. The eigenvalue problem can be rewritten as

A·v-·v=0
A·v-·I·v=0
(A-·I)·v=0

If v is non-zero, this equation will only have a solution if

|A-·I|=0

You can check that this matrix has the desired eigensystem.