Question

If A.x=\lambda x,where A=\begin{vmatrix}2&2&-2\\1&3&1\\1&2&2\end{vmatrix},determine the eigen values of the matrix A, and an eigen vector corresponding to each eigen value. If \lambda=2,what is b

Answer 1

Answer on question #41199 – Math – Linear Algebra

We have

$$A = \left( \begin{array}{ccc} 2 & 2 & -2 \\ 1 & 3 & 1 \\ 1 & 2 & 2 \end{array} \right)$$

To find the eigen values of this matrix we should find the following determinant and equate it to 0

$$\begin{aligned} |A - \lambda E| &= \left| \begin{array}{ccc} 2 - \lambda & 2 & -2 \\ 1 & 3 - \lambda & 1 \\ 1 & 2 & 2 - \lambda \end{array} \right| = (2 - \lambda)^2 (3 - \lambda) - 4 + 2 + 2(3 - \lambda) - 4(2 - \lambda) = 0 \\ &\quad - \lambda^3 + 7\lambda^2 - 14\lambda + 8 = 0 \\ &\quad (\lambda - 1)(\lambda - 2)(\lambda - 4) = 0 \\ &\quad \lambda = 1; 2; 4. \end{aligned}$$

The corresponding eigen vectors are

For $\lambda = 1$:

$$\begin{aligned} & \text{Ax} - \text{Ex} = 0 \\ & \left( \begin{array}{ccc} 1 & 2 & -2 \\ 1 & 2 & 1 \\ 1 & 2 & 1 \end{array} \right) x = 0 \\ &\left\{ \begin{array}{l} x_1 + 2x_2 - 2x_3 = 0 \\ x_1 + 2x_2 + x_3 = 0 \\ x_1 + 2x_2 + x_3 = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x_1 = -2x_2 + 2x_3 \\ -2x_2 + 2x_3 + 2x_2 + x_3 = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x_3 = 0 \\ x_1 = -2x_2 \end{array} \right. \end{aligned}$$

The eigen vector is $x = (-2; 1; 0)$.

For $\lambda = 2$:

$$\begin{aligned} & \text{Ax} - 2\text{Ex} = 0 \\ & \left( \begin{array}{ccc} 0 & 2 & -2 \\ 1 & 1 & 1 \\ 1 & 2 & 0 \end{array} \right) x = 0 \\ &\left\{ \begin{array}{l} 2x_2 - 2x_3 = 0 \\ x_1 + x_2 + x_3 = 0 \\ x_1 + 2x_2 = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{c} x_1 = -2x_2 \\ x_3 = x_2 \\ -2x_2 + x_2 + x_2 = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{c} x_1 = -2; \\ x_2 = 1; \\ x_3 = 1 \end{array} \right. \end{aligned}$$

The eigen vector is $x = (-2; 1; 1)$.

For $\lambda = 4$:

$$\begin{aligned} & \text{Ax} - 4\text{Ex} = 0 \\ &\left( \begin{array}{ccc} -2 & 2 & -2 \\ 1 & -1 & 1 \\ 1 & 2 & -2 \end{array} \right) x = 0 \\ &\left\{ \begin{array}{l} -2x_1 + 2x_2 - 2x_3 = 0 \\ x_1 - x_2 + x_3 = 0 \\ x_1 + 2x_2 - 2x_3 = 0 \end{array} \right. \Rightarrow \left\{ \begin{array}{l} x_1 = 0 \\ x_2 = 1 \\ x_3 = 1 \end{array} \right. \end{aligned}$$

The eigen vector is $x = (0; 1; 1)$.

http://www.AssignmentExpert.com