# Thread: which is non singular?

1. ## which is non singular?

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??

2) Let T : R -> R 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??

2. Originally Posted by poorna
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??
Yes, you are.

Tonio

3. oh yeah, R3-> R3