Question

the eigen values of a matrix A are 2,3,1 then find the eigen values of A^-1 + A^2​

Answer 1

Answer:

A=

⎣

⎢

⎢

⎡

2

2

−1

1

3

−1

1

4

−2

⎦

⎥

⎥

⎤

∣A−λI∣=0

⎣

⎢

⎢

⎡

2−λ

2

−1

1

3−λ

−1

1

4

−2−λ

⎦

⎥

⎥

⎤

=0

(2−λ)[(3−λ)(−2−λ)+4]−1[−4−2λ+4]+[−2+3−λ]=0

(2−λ)[λ

2

−λ−2]+λ+1=0

−λ

3

+3λ

2

+λ−3=0

λ

3

−3λ

2

−λ+3=0

(λ−1)(λ

2

−2λ−3)=0

(λ−1)(λ+1)(λ−3)=0

λ=−1,1,3

Therefore, Eigen values of matrix A are −1,1,3

Answer 2

Answer:

The eigen values of $A^{-1} +A^{2}$ is 4.5, 9.33, 2.

Step-by-step explanation:

Tip:

• According to Cayley Hamilton theorem, every square matrix satisfies its Characteristic equation.
• Putting the values of eigen value at place of A we can get Eigen values of   $A^{-1} +A^{2}$

Step1 of 1:

• Put A =2, then Eigen value is

         $= 2^{-1} +2^{2}\\=4.5$

• Put A =3 , then Eigen value is
•     $= 3^{-1} +3^{2}\\=9.33$

• Put A =1, then Eigen value is
•     $= 1^{-1} +1^{2}\\=2$