Not very sure where to start with this question. Can someone point me in the right direction?
2026-03-28 04:01:49.1774670509
Find a 2x2 Matrix A where Eλ is the eigenspace associated with the eigenvalue λ
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Start with $$ A=\left(\begin{array}{cc} a&b \\ c&d\end{array}\right) $$ and search for the unknowns $a,b,c,d$ so that \begin{align} A\left(\begin{array}{c} -2 \\ 5\end{array}\right)&=5\left(\begin{array}{c} -2 \\ 5\end{array}\right)\tag{1} \\ A\left(\begin{array}{c} 1 \\ -3\end{array}\right)&=1\left(\begin{array}{c} 1 \\ -3\end{array}\right)\, .\tag{2} \end{align} You will get $4$ equations for your $4$ unknowns, and this system has a unique solution. As a bonus you can verify that $\hbox{Tr}(A)=a+d=5+1=6$ and that $\hbox{det}(A)=5\times 1=5$. As a hint, the entries of $A$ are all integers.