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xz_007 [3.2K]
2 years ago
11

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 th

e corresponding eigenspace. The eigen value = has multiplicity = and the dimension of the corresponding eigenspace is:_______.
Mathematics
1 answer:
Aliun [14]2 years ago
3 0

Answer:

a) -4

b) 1

c) 1

Step-by-step 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\sqrt{3}\\\\\lambda_3=-1-i\sqrt{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)

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