Hello.

I have two doubts

Let A be a 3 x 3 matrix with eigen values 1,-1,3. then which is the correct option?

a) A^2 + A is non-singular

b) A^2 - A is non-singular

c) A^2 + 3A is non-singular

d) A^2 -3A is non-singular

A is equivalent to the diagonal matrix having diagonal having diag entries 1,-1,3

So A^2 s equivalent to the diagonal matrix having diagonal having diag entries 1,1,9.

So I think the answer is c) A^2 + 3A is non-singular. Am I right??

Yes. You can also see this by checking the characteristic polynomial of A, $\displaystyle p_A(x)=(x-1)(x+1)(x-3) $ This blows options (a),(b),(d) , so only (c) is left...
2) Let T : R -> R

According to the continuation of the question , this should be $\displaystyle T\,:\,\mathbb{R}^3\to \mathbb{R}^3$ , isn't it?
be a linear transformation and I be the identity transformation of R3. If there is a scalar c and a non zero vector x in R3 such that T(x)=cx, then the rank of (T-cI) cannot be

a) 0

b) 1

C) 2

D) 3

c is the eigen calue of the transformation T. so T-cI is singular. So rank (T-cI) is not 3.

Am i right??