307 Construct 3 different 2x2 matrices

[karush]

Construct 3 different $2\times2$ matrices
1. having two distinct real eigenvalues,
$A=\begin{bmatrix}
2 & -1\\
-1& 2
\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & -1 \\ -1 & 2 - \lambda \end{array} \right|=\lambda^{2} - 4 \lambda + 3 \therefore \lambda_1=3\ \lambda_2=1$

2. a pair of complex eigenvalues.
$\left| \begin{array}{cc}
2 - \lambda & 1 \\
-1 & 2 - \lambda
\end{array} \right|=\lambda^{2} - 4 \lambda + 5\quad \lambda_{1}=2 - i,\lambda_{1}=2 + i$

3. two identical real eigenvalues,

ok hopefully 1 and 2 are ok

on 3 I was just going to do this $(\lambda-2)(\lambda-2)$
and go backwards to a matrix but ?

 

[jvicens]

For the third matrix, you can use the following:
$B=\begin{bmatrix}
2 & 0\\
0 & 2
\end{bmatrix},\quad \left| \begin{array}{rr} 2 - \lambda & 0 \\ 0 & 2 - \lambda \end{array} \right|=(2-\lambda)^2=0 \therefore \lambda_1=\lambda_2=2$.
This matrix has two identical real eigenvalues of 2.