Solve Eigenvector Problem: Find Eigenvalues & Eigenvectors

[geft]

Homework Statement

Find the eigenvalues and eigenvectors of $$\left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 1 & 1 & 0 \end{array} \right)$$

Homework Equations

$$p(\lambda) = det(A - \lambda I) = 0$$

The Attempt at a Solution

$$A - \lambda I = \left( \begin{array}{ccc} 2-\lambda & 0 & 0 \\ 0 & 3-\lambda & 4 \\ 1 & 1 & 0 \end{array} \right)$$

$$det(A - \lambda I) = (2 - \lambda)(-4) + 1 = 0$$

$$-8 + 4 \lambda + 1 = 0$$

$$4\lambda = 7$$

$$\lambda = \frac{4}{7}$$

$$A - \lambda I = \left( \begin{array}{ccc} \frac{1}{4} & 0 & 0 \\ 0 & \frac{5}{7} & 4 \\ 1 & 1 & 0 \end{array} \right)$$

$$\frac{1}{4}x_1 = 0$$

$$\frac{5}{4}x_2 + 4x_3 = 0$$

Eigenvector = $$\left( \begin{array}{ccc} 0 \\ 0 \\ 0 \end{array} \right)$$

 

[Mark44]

This should be in the Calculus and Beyond section.

For your formatting problems, use [ tex ] and [ /tex ] tags (without the extra spaces I put in, instead of the inline LaTeX tags, [ itex ], you used.

 

[geft]

Thanks. Should I repost this in the other section?

 

[Pengwuino]

As for the problem, in your $$A-\lambda I$$ matrix, you still have a $$-\lambda$$ in the 3rd row, 3rd column entry. It isn't simply 0.

 

[geft]

As for the problem, in your $$A-\lambda I$$ matrix, you still have a $$-\lambda$$ in the 3rd row, 3rd column entry. It isn't simply 0.

Oh, you're right. Thanks!