Question

        [ 1 0 -1

3. Consider the matrix A =  0 3 0

                      -1 0 1 ]

1. Find the eigenvalues of A.
2. Find the eigenspaces corresponding to each eigenvalue from A.

Answer 1

$A=\begin{pmatrix} 1& 0&-1 \\ 0 & 3&0\\-1&0&1 \end{pmatrix}$

$|λI-A|=\begin{vmatrix} λ-1& 0&1 \\ 0 & λ-3&0\\1&0&λ-1 \end{vmatrix} = 0$

$=>|λI-A|=(λ-3)[(λ-1)²-1)]=0$

$=>|λI-A|=(λ-3)(λ²-2λ)=λ(λ-3)(λ-2)=0$

$∴ λ=0,3,2$

when $λ=0$

$MX=\begin{pmatrix} -1& 0&1 \\ 0 & -3&0\\1&0&-1 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix} =\begin{pmatrix} 0 \\ 0\\0 \end{pmatrix}$

$=>\begin{pmatrix} -1& 0&1 \\ 0 & -3&0\\1&0&-1 \end{pmatrix}\backsim\begin{pmatrix} 1&0&-1 \\0&-3&0 \\-1&0&1 \end{pmatrix}\backsim\begin{pmatrix} 1&0&-1 \\ 0&-3&0\\0&0&0 \end{pmatrix}\backsim\begin{pmatrix} 1&0&-1 \\ 0&1&0\\0&0&0 \end{pmatrix}$

$=>x_{1}-x_{3}=0$ $x_{2}=0$

Let $x_{3}=t$

$x_{1}=x_{3}=t,$ $x_{2}=0$

The eigenspace corresponding to the eigenvalue λ=0 is

$\{\begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix}∈ℝ³ : x_{1}=x₃, x₂=0\}$

when $λ=2$

$MX=\begin{pmatrix} 1& 0&1 \\ 0 & -1&0\\1&0&1 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix} =\begin{pmatrix} 0 \\ 0\\0 \end{pmatrix}$

$=>\begin{pmatrix} 1& 0&1 \\ 0 & -1&0\\1&0&1 \end{pmatrix}\backsim\begin{pmatrix} 1&0&1 \\0&-1&0 \\0&0&0 \end{pmatrix}\backsim\begin{pmatrix} 1&0&1 \\ 0&1&0\\0&0&0 \end{pmatrix}$

$=>x_{1}+x_{3}=0$ , $x_{2}=0$

The eigenspace corresponding to the eigenvalue λ=2 is

$\{\begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix}∈ℝ³ : x_{1}+x₃=x₂=0\}$

when $λ=3$

$MX=\begin{pmatrix} 2& 0&1 \\ 0 & 0&0\\1&0&2 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix} =\begin{pmatrix} 0 \\ 0\\0 \end{pmatrix}$

$=>\begin{pmatrix} 2& 0&1 \\ 0 & 0&0\\1&0&2 \end{pmatrix}\backsim\begin{pmatrix} 1&0&2 \\2&0&1 \\0&0&0 \end{pmatrix}\backsim\begin{pmatrix} 1&0&1 \\ 0&0&-3\\0&0&0 \end{pmatrix}\backsim\begin{pmatrix} 1&0&1 \\ 0&0&1\\0&0&0 \end{pmatrix}$

$=>x_{1}+x_{3}=0$ , $x_{3}=0$

$∴x_{1}=0$

The eigenspace corresponding to the eigenvalue λ=3 is

$\{\begin{pmatrix} x_{1} \\ x_{2}\\x_{3} \end{pmatrix}∈ℝ³ : x_{1}=0=x₃\}$