GATE Mechanical 2021 Set 2 | Question: 1

Question

Consider an $n \times n$ matrix $\text{A}$ and a non-zero $n \times 1$ vector $p.$ Their product $Ap=\alpha ^{2}p$, where $\alpha \in \Re$ and $\alpha \notin  \left \{ -1,0,1 \right \}$. Based on the given information, the eigen value of $A^{2}$ is:

• A. $\alpha$
• B. $\alpha ^{2}$
• C. $\surd{\alpha }$
• D. $\alpha ^{4}$

Answer 1

Here the Equation is given as 

Ap=(α^2)p

Here p is a non zero vector 

A is the matrix

We are linearly transforming(Multiplying a Matrix with a Vector is Linear Transformation) a Non zero Vector By a Matrix A 

The Changing is Same as multiplying that Vector with a Scalar 

So, this is Straight enough that (α^2) is eigen value of A 

And We know if k is a eigen value of A 

Then K^2 will be the eigen value of A^2

Similarly Here Also (α^4) would be the eigen value of A^2