solved find eigen values of the matrix [5 4;1 2]

Question : Find eigen values of the matrix [5 4;1 2]

Solution :

Given matrix \[A=\left(\begin{matrix}5&4\\1&2\\\end{matrix}\right)\] eigen values of A are given by \[\left|A-\lambda I\right|=0\] Where I is an identity matrix of order 2. \[\left|\left(\begin{matrix}5&4\\1&2\\\end{matrix}\right)-\lambda\left(\begin{matrix}1&0\\0&1\\\end{matrix}\right)\right|=0\] \[\left|\left(\begin{matrix}5-\lambda&4\\1&2-\lambda\\\end{matrix}\right)\right|=0\] \[\left(5-\lambda\right)\left(2-\lambda\right)-4=0\] \[6-7\lambda+\lambda^2=0\] \[\left(\lambda-6\right)\left(\lambda-1\right)=0\] \[\lambda=\ 6,\ 1 \]