Finding Eigenspaces of a Matrix

[Dustinsfl]

\begin{bmatrix}<br /> 3 &amp; 2\\<br /> 4 &amp; 1<br /> \end{bmatrix}
det(A-\lambda I)=\begin{vmatrix}<br /> 3-\lambda &amp; 2\\<br /> 4 &amp; 1-\lambda<br /> \end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5
\lambda_{1}=5 and \lambda_{2}=-1
When \lambda=5, \begin{bmatrix}<br /> -2 &amp; 2\\<br /> 4 &amp; -4<br /> \end{bmatrix}\Rightarrow \begin{bmatrix}<br /> 1 &amp; -1\\<br /> 0 &amp; 0<br /> \end{bmatrix}
The eigenspace for \lambda_{1} is \begin{bmatrix}<br /> 1\\<br /> 1<br /> \end{bmatrix}
When \lambda=-1, \begin{bmatrix}<br /> 4 &amp; 2\\<br /> 4 &amp; 2<br /> \end{bmatrix}\Rightarrow \begin{bmatrix}<br /> 1 &amp; 0\\<br /> 0 &amp; 1<br /> \end{bmatrix}
The eigenspace for\lambda_{2} is \begin{bmatrix}<br /> 0\\<br /> 0<br /> \end{bmatrix}

I don't know what is going wrong but my second Eigenspace is wrong compared to the books answer which is \begin{bmatrix}<br /> 1\\<br /> -2<br /> \end{bmatrix}

 

[Mark44]

\begin{bmatrix}<br /> 3 &amp; 2\\<br /> 4 &amp; 1<br /> \end{bmatrix}
det(A-\lambda I)=\begin{vmatrix}<br /> 3-\lambda &amp; 2\\<br /> 4 &amp; 1-\lambda<br /> \end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5
\lambda_{1}=5 and \lambda_{2}=-1
When \lambda=5, \begin{bmatrix}<br /> -2 &amp; 2\\<br /> 4 &amp; -4<br /> \end{bmatrix}\Rightarrow \begin{bmatrix}<br /> 1 &amp; -1\\<br /> 0 &amp; 0<br /> \end{bmatrix}
The eigenspace for \lambda_{1} is \begin{bmatrix}<br /> 1\\<br /> 1<br /> \end{bmatrix}
When \lambda=-1, \begin{bmatrix}<br /> 4 &amp; 2\\<br /> 4 &amp; 2<br /> \end{bmatrix}\Rightarrow \begin{bmatrix}<br /> 1 &amp; 0\\<br /> 0 &amp; 1<br /> \end{bmatrix}

Your mistake is above. The [4 2; 4 2] matrix doesn't row reduce to the identity matrix. Try again.
Each matrix for calculating the eigenspace can't reduce to the identity; otherwise its determinant would not be zero.

The eigenspace for\lambda_{2} is \begin{bmatrix}<br /> 0\\<br /> 0<br /> \end{bmatrix}

I don't know what is going wrong but my second Eigenspace is wrong compared to the books answer which is \begin{bmatrix}<br /> 1\\<br /> -2<br /> \end{bmatrix}

 

[Dustinsfl]

I had a -2 entered into my calc.