Question

.3 Suppose the eigenvalues of a 3x3 matrix A are A, 4, , and A 6' %3D with corresponding eigenvectors v,= V2= and v Let -2 -5 6. 11 Find the solution of the equation x Ax, for the specified x, and describe what happens ask-o. 13 Find the solution of the equation X1AX Choose the correct answer below. 4. 1. O A. X=2.(4)* +3. -4 1. 6. -5 -2 -3 O B. X=2.(4)* 0 +3. 1. -5 6.

Answer 1

Here, the solution of the equation $x_{k+1}=Ax_k$ is :

$x_k=2*(4)^k\begin{bmatrix} 1\\ 0\\ -2 \end{bmatrix}+3*\left ( \frac{5}{6} \right )^k\begin{bmatrix} 4\\ 1\\ -5 \end{bmatrix}+\left ( \frac{1}{6} \right )^k\begin{bmatrix} -3\\ -4\\ 6 \end{bmatrix}$

When $k\to\infty$ , then we have,

$x_k\approx2*(4)^k\begin{bmatrix} 1\\ 0\\ -2 \end{bmatrix}+3*0*\begin{bmatrix} 4\\ 1\\ -5 \end{bmatrix}+0*\begin{bmatrix} -3\\ -4\\ 6 \end{bmatrix}$

i.e., $x_k\approx2*(4)^k\begin{bmatrix} 1\\ 0\\ -2 \end{bmatrix}+0+0$

i.e., $x_k\approx2*(4)^k\begin{bmatrix} 1\\ 0\\ -2 \end{bmatrix}$