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
A
\(\begin{pmatrix}6 & 3\\3 & 20\end{pmatrix}\)
B
\(\begin{pmatrix}9 & 8\\22 & 23\end{pmatrix}\)
C
\(\begin{pmatrix}10 & 7\\21 & 24\end{pmatrix}\)
D
\(\begin{pmatrix}7 & 2\\12 & 21\end{pmatrix}\)
Verified Correct Answer: C.
Step-by-Step Mathematical Solution
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 = \(\begin{pmatrix}7 & 6\\18 & 19\end{pmatrix}\) + \(\begin{pmatrix}2 & 1\\3 & 4\end{pmatrix}\) + \(\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\)
= \(\begin{pmatrix}10 & 7\\21 & 24\end{pmatrix}\)
Source: Myschool Past Questions Archive (Record #70080)