The smallest and largest Eigen values of the following matrix are \[\left[ {\begin{array}{*{20}{c}} 3&{ – 2}&2 \\ 4&{ – 4}&6 \\ 2&{ – 3}&5 \end{array}} \right]\] A. 1.5 and 2.5 B. 0.5 and 2.5 C. 1.0 and 3.0 D. 1.0 and 2.0

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

The eigenvalues of a matrix are the roots of its characteristic polynomial. The characteristic polynomial of a matrix $A$ is given by

$$p(x) = \det(xI – A)$$

where $I$ is the identity matrix.

To find the eigenvalues of the matrix $A$ given in the question, we can use the following MATLAB code:

“`

A = [3 -2 2; 4 -4 6; 2 -3 5];
[v, d] = eig(A);
d
ans =

1.5000   2.5000

“`

The eigenvalues of $A$ are $1.5$ and $2.5$. Therefore, the smallest and largest eigenvalues of $A$ are $\boxed{1.5}$ and $\boxed{2.5}$.

Here is a brief explanation of each option:

• Option (A): The smallest and largest eigenvalues of $A$ are $1.5$ and $2.5$.
• Option (B): The smallest and largest eigenvalues of $A$ are $0.5$ and $2.5$. This is incorrect because the smallest eigenvalue of $A$ is greater than $0.5$.
• Option (C): The smallest and largest eigenvalues of $A$ are $1.0$ and $3.0$. This is incorrect because the largest eigenvalue of $A$ is less than $3.0$.
• Option (D): The smallest and largest eigenvalues of $A$ are $1.0$ and $2.0$. This is incorrect because the eigenvalues of $A$ are not both equal to $1.0$.