Question

for a matrix a of order 3 and |A| = 32 and two of the Eigen values are 8 and 2 , find the sum of the Eigen values​

Answer 1

Step-by-step explanation:

Given, A=

⎣

⎢

⎢

⎡

2

2

−1

1

3

−1

1

4

−2

⎦

⎥

⎥

⎤

The characteristic equation of A is given by

∣A−λI∣=0

⇒

∣

∣

∣

∣

∣

∣

∣

∣

2−λ

2

−1

1

3−λ

−1

1

4

−2−λ

∣

∣

∣

∣

∣

∣

∣

∣

=0

⇒λ

3

−3λ

2

−λ+3=0

Here, λ=1 satisfies the equation

⇒(λ−1)(λ

2

−2λ−3)=0

⇒(λ−1)(λ(λ−3)+1(λ−3))=0

⇒(λ−1)(λ−3)(λ+1)=0

⇒λ=1,−1,3

Hence, the eigen values are 1,−1,3

Answer 2

Answer:

Sum of the eigen values is 12.

Step-by-step explanation:

Given two eigen values of a matrix A of order 3 are 8 and 2.

Determinant of matrix A, detA = 32.

Let the third eigen value = x

We know that the product of eigen values of a square matrix is equal to the determinant of that matrix.

So, 8 × 2 × x = 32

     16 × x = 32

     x = 2

The third eigen value is 2.

Thus, the sum of the eigen values is = 8 + 2 + 2 = 12.