Question

Consider the system of equation $A_{(}n\times n)X_{(}n\times I)={\lambda }_{(}n\times I)$ where $lambda$ is a scalar. Let $({\lambda }_i,X_i)$ be an eige npari of an eigen value and its corresponding eigen vector for real matrix $A$. let $I$ be a $n\times n$ unit matrix. Which one of the followign statements is not correct?

• A. For a homogeneous $n\times n$ system of linear equation, $(A-\lambda I)x=0$ having a non-trivial solution, the rank of $(A-\lambda I)$is less than n
• B. For matrix $A^m$, $m$ being a positve integer, $({\lambda }_i^m,X_i^m)$ will be the eigen pair for all $i$
• C. If $A^T=A^{-1}$ thne $\left[{\lambda }_i\right]=1$ for all $i$
• D. If $A^T=A$ thne ${\lambda }_i$ is real for all $i$

Answer 1

The correct option is B For matrix $A^m$, $m$ being a positve integer, $({\lambda }_i^m,X_i^m)$ will be the eigen pair for all $i$

If $\lambda$ is an eigan value of matrix $A$ hten eigen value of $A^n$ is ${\lambda }^n$.
if $X$ is an eigen vector of matrix $A$ then eigen vector of $A^n$ is $X$.