Problem : How to find eigen values of 4 by 4 matrix ? Answer : For the given matrix\[A=\left(\begin{matrix}2&0&1&1\\1&2&5&-5\\0&0&3&0\\0&0&1&3\\\end{matrix}\right)\]Eigen values :Eigen values k of the matrix A are calculated by \[\left|A-kI\right|=0\]Where I is an identity matrix of order 4.\[=>\left|\begin{matrix}2-k&0&1&1\\1&2-k&5&-5\\0&0&3-k&0\\0&0&1&3-k\\\end{matrix}\right|=0\] \[=>\left(2-k\right)\left|\begin{matrix}2-k&5&-5\\0&3-k&0\\0&1&3-k\\\end{matrix}\right|-0\left|\begin{matrix}1&5&-5\\0&3-k&0\\0&1&3-k\\\end{matrix}\right|+1\left|\begin{matrix}1&2-k&-5\\0&0&0\\0&0&3-k\\\end{matrix}\right|-\left|\begin{matrix}1&2-k&5\\0&0&3-k\\0&0&1\\\end{matrix}\right|=0\]\[=>\left(2-k\right)\left[\left(2-k\right)\left(\left(3-k\right)^2\right)-5\left(0\right)+\left(-5\right)\left(0\right)\right]-0+1\left[1\left(0\right)-\left(2-k\right)\left(0\right)-5\left(0\right)\right]=0\]…
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