Question

Consider the matrix $\left[\begin{array}{cc}5& -1\\ 4& 1\end{array}\right]$ which one of the following statements is TRUE for the eigen values and eigen vectors of the matrix?

• A. Eigen value 3 has a multiplicity of 2 and only one independent eigen vector exists
• B. Eigen value 3 has a multiplicity of 2 and two independent eigen vectors exist
• C. Eigen value 3 has a multiplicity of 2 and no independent eigent vector exists
• D. Eigen values are 3 and -3 and two independent eighen vectors exist.

Answer 1

The correct option is A Eigen value 3 has a multiplicity of 2 and only one independent eigen vector exists

[A] = $\left[\begin{array}{cc}5& -1\\ 4& 1\end{array}\right]$

For eigen value

$|A-\lambda I|$ = $\left|\begin{array}{cc}5-\lambda & -1\\ 4& 1-\lambda \end{array}\right|$ = 0

$\Rightarrow$ $(5-\lambda )(1-\lambda )$+4 = 0

$5-5\lambda -\lambda +{\lambda }^2+4$ = 0

${\lambda }^2-6\lambda$+9 = 0

$(\lambda -3{)}^2$ = 0

$\lambda$ = 3, 3

So it has multiplicity 'two'

For eigen vector

$[A-\lambda I]\left[\begin{array}{c}x\\ y\end{array}\right]$ = 0

$\left[\begin{array}{cc}5-3& -1\\ 4& 1-3\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$

2x - y = 0 $\Rightarrow$ y = 2x.

Let x = k, then y = 2k

So X = $\left[\begin{array}{c}x\\ y\end{array}\right]$ = $\left[\begin{array}{c}k\\ 2k\end{array}\right]$ $\sim$$\left[\begin{array}{c}1\\ 2\end{array}\right]$

So, only one independent eigen vector exists for $\lambda$ = 3.