Here we have to recall the property :
a) If a matrix A is modified A = f(A) then the individual eigen values will also be updated in the same manner
b) The product of eigen values will give us the determinant of the matrix.
c) If the determinant of the matrix is non zero then it is said to be invertible matrix.
d) In an n * n matrix , at most n distinct eigen values possible.
In this context a) point means ,
If say x is the eigen value of A , then the corresponding eigen value
for A2 + 3A will be : x2 + 3x
So given that one of the eigen values of A2 + 3A is 18..So using a) property mentioned above , we have :
x2 + 3x = 18
==> x = 3
So 3 is the corresponding eigen value of original matrix..And two of the eigen values already given : 1 , -1
So determinant of matrix A = 3 * 1 * -1
= -3 which is non zero
Hence A is invertible..
Now for A2 + A matrix we need to calculate other 2 eigen values which can be found using the eigen values 1 and -1 of A.
So eigen value of the new matrix corresponding to the eigen value 1 = 12 + 3(1) = 4
eigen value of the new matrix corresponding to the eigen value -1 = (-1)2 + 3(-1) = 1 - 3 = -2
So product of eigen values of new matrix = 18 * 4 * -2 = -144
Hence determinant of new matrix = -144 which is non zero as well
So the new matrix is also invertible.
Hence A is invertible as well as A2 + 3A is invertible.
Hence A) is the correct option.