133 - FML 3 - 18.1W e.g. 18

Properties

4 Activities

 If   ⁣A‾\bcb{\ul{A}}A​ is a 4×4\bcb{4 \times 4}4×4 matrix with eigenvalues λ1=λ2=λ3=1\bcb{\lambda_1 =  \lambda_2 = \lambda_3 = 1}λ1​=λ2​=λ3​=1 and λ4=2\bcb{\lambda_4 = 2}λ4​=2, which of the following statements are true?

(a) The homogeneous system of linear equations associated with   ⁣A‾\bcb{\ul{A}}A​ has only the trivial solution.

B) trace(  ⁣A‾)=5\bcb{\text{trace}(\ul{A}) = 5}trace(A​)=5. 

C) The geometric multiplicity of λ=1\bcb{\lambda = 1}λ=1 is 3\bcb{3}3.

D) The matrix   ⁣M‾=  ⁣A‾−2  ⁣I‾\bcb{\ul{M} = \ul{A} - 2\ul{I}}M​=A​−2I​   is not invertible. 

I don't know

133 - FML 3 - 18.1W e.g. 19

 If   ⁣A‾\bcb{\ul{A}}A​ be a 3×3\bcb{3 \times 3}3×3 matrix where trace(  ⁣A‾)=−1\bcb{\text{trace}(\ul{A}) = -1}trace(A​)=−1 and det(  ⁣A‾)=−1\bcb{\det{\ul{A}} =  -1}det(A​)=−1, then the possible eigenvalues of   ⁣A‾\bcb{\ul{A}}A​ are:

133 - FML 3 - 18.1W e.g. 70.5

 Suppose 5\bcb{5}5 and −2\bcb{-2}−2 are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}A​, which of the following statements must be true?
		det(  ⁣A‾−5  ⁣I‾)=det(  ⁣A‾+2  ⁣I‾)\bcb{\det{\ul{A} - 5\ul{I}} = \det{\ul{A} + 2 \ul{I}}}det(A​−5I​)=det(A​+2I​). 

133 - FML 3 - 18.1W e.g. 70.4

 Suppose 5\bcb{5}5 and −2\bcb{-2}−2 are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}A​, which of the following statements must be true?
		trace(  ⁣A‾−5  ⁣I‾)=trace(  ⁣A‾+2  ⁣I‾)\bcb{\text{trace}(\ul{A} - 5\ul{I}) = \text{trace}(\ul{A} + 2\ul{I})}trace(A​−5I​)=trace(A​+2I​).

133 - FML 3 - 18.1W e.g. 70.3

 Suppose 5\bcb{5}5 and −2\bcb{-2}−2 are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}A​, which of the following statements must be true?
		det(  ⁣A‾−5  ⁣I‾)=0\bcb{\det{\ul{A} - 5\ul{I}} = 0}det(A​−5I​)=0, and   ⁣A‾v⃗=−2v⃗\bcb{\ul{A} \ol{v} = -2 \ol{v}}A​v=−2v has only the trivial solution.

133 - FML 3 - 18.1W e.g. 70.2

 Suppose 5\bcb{5}5 and −2\bcb{-2}−2 are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}A​, which of the following statements must be true?
		A−5I\bcb{A-5I}A−5I is invertible and det(  ⁣A‾+2  ⁣I‾)=0\bcb{\det{\ul{A} + 2\ul{I} } = 0}det(A​+2I​)=0.

133 - FML 3 - 18.1W e.g. 70.1

 Suppose 5\bcb{5}5 and −2\bcb{-2}−2 are eigenvalues of the matrix   ⁣A‾\bcb{\ul{A}}A​, which of the following statements must be true?
		A−5I\bcb{A - 5I}A−5I is not invertible and there is a nontrivial solution to   ⁣A‾v⃗=−2v⃗\bcb{\ul{A} \ol{v} = -2 \ol{v}}A​v=−2v.

Properties

Practice: Similar Matrices
Consider the matrix A=[−1421]A=
\left[
\begin{array}{rrr}
-1 & 4\\
2 & 1
\end{array}
\right]A=[−12​41​].

Practice Question: Eigenvalues of Triangular Matrices

Prove that any element on the main diagonal of a triangular matrix is an eigenvalue.

133 - FML 3 - 18.1W e.g. 18

Related Topics

Properties

If $\bcb{\ul{A}}$ is a $\bcb{4 \times 4}$ matrix with eigenvalues $\bcb{\lambda_1 = \lambda_2 = \lambda_3 = 1}$ and $\bcb{\lambda_4 = 2}$ , which of the following statements are true?

More Properties Questions:

133 - FML 3 - 18.1W e.g. 19

133 - FML 3 - 18.1W e.g. 70.5

133 - FML 3 - 18.1W e.g. 70.4

133 - FML 3 - 18.1W e.g. 70.3

133 - FML 3 - 18.1W e.g. 70.2

133 - FML 3 - 18.1W e.g. 70.1

Properties

Practice Question: Eigenvalues of Triangular Matrices

133 - FML 3 - 18.1W e.g. 18

Related Topics

Properties

If ⁣A‾\bcb{\ul{A}}A​ is a 4×4\bcb{4 \times 4}4×4 matrix with eigenvalues λ1=λ2=λ3=1\bcb{\lambda_1 = \lambda_2 = \lambda_3 = 1}λ1​=λ2​=λ3​=1 and λ4=2\bcb{\lambda_4 = 2}λ4​=2, which of the following statements are true?

More Properties Questions:

133 - FML 3 - 18.1W e.g. 19

133 - FML 3 - 18.1W e.g. 70.5

133 - FML 3 - 18.1W e.g. 70.4

133 - FML 3 - 18.1W e.g. 70.3

133 - FML 3 - 18.1W e.g. 70.2

133 - FML 3 - 18.1W e.g. 70.1

Properties

Practice Question: Eigenvalues of Triangular Matrices

If $\bcb{\ul{A}}$ is a $\bcb{4 \times 4}$ matrix with eigenvalues $\bcb{\lambda_1 = \lambda_2 = \lambda_3 = 1}$ and $\bcb{\lambda_4 = 2}$ , which of the following statements are true?