Question

The matrix A= (−3 0 1, 2 −4 2, −3 −2 1) has one real eigenvalue. Find this eigenvalue, its multiplicity, and the dimension of the corresponding eigenspace. The eigen value = has multiplicity = and the dimension of the corresponding eigenspace is:_______.

Answer 1

a) -4

b) 1

c) 1

Explanation:

a) The matrix A is given by:

`A=\left[\begin{array}{ccc}-3&0&1\\2&-4&2\\-3&-2&1\end{array}\right]`

to find the eigenvalues of the matrix you use the following:

`det(A-\lambda I)=0`

where lambda are the eigenvalues and I is the identity matrix. By replacing you obtain:

`A-\lambda I=\left[\begin{array}{ccc}-3-\lambda&0&1\\2&-4-\lambda&2\\-3&-2&1-\lambda\end{array}\right]`

and by taking the determinant:

`[(-3-\lambda)(-4-\lambda)(1-\lambda)+(0)(2)(-3)+(2)(-2)(1)]-[(1)(-4-\lambda)(-3)+(0)(2)(1-\lambda)+(2)(-2)(-3-\lambda)]=0\\\\-\lambda^3-6\lambda^2-12\lambda-16=0`

and the roots of this polynomial is:

`\lambda_1=-4\\\\\lambda_2=-1+i√(3)\\\\\lambda_3=-1-i√(3)`

hence, the real eigenvalue of the matrix A is -4.

b) The multiplicity of the eigenvalue is 1.

c) The dimension of the eigenspace is 1 (because the multiplicity determines the dimension of the eigenspace)