The lowest Eigen value of the 2 × 2 matrix \[\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&3 \end{array}} \right]\] A. 1 B. 2 C. 3 D. 5

The correct answer is $\boxed{\text{A}}$.

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a 2×2 matrix can be written as:

$$p(x) = |A – xI|$$

where $A$ is the matrix and $I$ is the identity matrix.

In this case, we have:

$$p(x) = | \begin{array}{cc} 4 & 2 \ 1 & 3 \end{array} – x \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} | = x^2 – 7x + 10$$

The eigenvalues of $A$ are the roots of $p(x)$, which are $x = 3$ and $x = 4$. Therefore, the lowest eigenvalue of $A$ is $\boxed{3}$.

To explain each option in brief:

• Option A: $1$ is not an eigenvalue of $A$. This can be verified by substituting $x = 1$ into the characteristic polynomial $p(x)$ and evaluating.
• Option B: $2$ is an eigenvalue of $A$. This can be verified by substituting $x = 2$ into the characteristic polynomial $p(x)$ and evaluating.
• Option C: $3$ is an eigenvalue of $A$. This has already been explained above.
• Option D: $5$ is not an eigenvalue of $A$. This can be verified by substituting $x = 5$ into the characteristic polynomial $p(x)$ and evaluating.