Given the matrix K = \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\), the matrix K\(^2\) + K + I, where I is the 2 x 2 identity matrix, is
K = \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\)
K\(^2\) = K = \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\) x K = \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\) = \(\begin{pmatrix}7 & 6\\18 & 19\end{pmatrix}\)
I = \(\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\)
K\(^2\) + K + I = \(\begin{pmatrix}7 & 6\\18 & 19\end{pmatrix}\) + \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\) + \(\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\)
= \(\begin{pmatrix}10 & 7\\21 & 24\end{pmatrix}\)
There is an explanation video available below.
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