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Given -1 and 1 are two eigen values of matrix A

and 18 is the eigen value of A^2 + 3A

so other eigen values of A^2 +  3A are 4 , -2

and other eigen value of A is 3 ( 3^2 + 3 )

Eigen value of A is -1 ,1, 3 and A^2 + 3A is 4,-2,18

No eigen value is 0 so both are invertible
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Let 3rd eigen value is x.

Eigen values of A2 +3A are 1+3, 1-3, x2 + 3x i.e 4, -2, x2 + 3x

Compare this with given condition. x2 + 3x = 18, x = -6, 3

None of eigen value is zero that means both A as well as A2 + 3A are invertible.

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