The eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&2 \\ 0&2 \end{array}} \right]\] are written in the form \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{a}} \end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}} 1 \\ {\text{b}} \end{array}} \right].\] What is a + b = ? A. 0 B. \[\frac{1}{2}\] C. 1 D. 2

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The correct answer is $\boxed{1}$.

The eigenvalues of the matrix $\left[ {\begin{array}{{20}{c}} 1&2 \ 0&2 \end{array}} \right]$ are $\lambda = 2$ and $\lambda = 0$. The eigenvectors corresponding to these eigenvalues are $\left[ {\begin{array}{{20}{c}} 1 \ 1 \end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}} 1 \ -1 \end{array}} \right]$, respectively. Therefore, $a + b = 1 + (-1) = \boxed{1}$.

Here is a more detailed explanation of the solution:

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of the matrix $\left[ {\begin{array}{*{20}{c}} 1&2 \ 0&2 \end{array}} \right]$ is $p(\lambda) = \lambda^2 – 2\lambda = (\lambda – 2)^2$. Therefore, the eigenvalues of the matrix are $\lambda = 2$ and $\lambda = 0$.

The eigenvectors of a matrix are the non-zero vectors that satisfy the equation $Av = \lambda v$ for some eigenvalue $\lambda$. The eigenvectors corresponding to the eigenvalues $\lambda = 2$ and $\lambda = 0$ of the matrix $\left[ {\begin{array}{{20}{c}} 1&2 \ 0&2 \end{array}} \right]$ are $\left[ {\begin{array}{{20}{c}} 1 \ 1 \end{array}} \right]$ and $\left[ {\begin{array}{*{20}{c}} 1 \ -1 \end{array}} \right]$, respectively.

Therefore, $a + b = 1 + (-1) = \boxed{1}$.