Question

The number of linearly independent eigen vectors of $\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$ is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is B $1$

eigen values: $|A-\lambda I|=0$
$\left|\begin{array}{cc}2-\lambda & 1\\ 0& 2-\lambda \end{array}\right|=0$
$\Rightarrow \lambda =2,2$ (Repeated)
Now, $A-2I=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$
$\because \rho (A-2I)=1$ & Nullity = Order - Rank $=2-1=1$
Hence, for $\lambda =2,$ only one LI Eigen vector.