Question

[ \left[begin{array}{rr} -1 & -frac{3}{2} frac{1}{2} & 1 end{array}right] ]

(a) the characteristic equation
(b) the eigenvalues (Enter your answers from smallest to largest.)

Answer 1

To find the characteristic equation and eigenvalues of the given matrix, we need to find the determinant of the matrix A - xI, where A is the given matrix and I is the identity matrix. The characteristic equation can be solved to obtain the eigenvalues.

Step-by-step explanation:

The given matrix is:

[[-1, -3/2, 1]]

To find the characteristic equation, we need to find the determinant of the matrix A - xI, where A is the given matrix and I is the identity matrix. The characteristic equation is given by det(A - xI) = 0. Evaluating the determinant:

[[-1 - x, -3/2, 1],[-1/2, -1 - x, 1],[0, 0, -1 - x]]

Expanding the determinant, we get:

(-1 - x)((-1 - x)(-1 - x) - 0) - (-3/2)(-1/2 - 0) = 0

Simplifying the equation:

(-1 - x)(x^2 + 2x + 1) + 3/2 = 0

Simplifying further:

-x^3 - 4x^2 - x - x^2 - 2x - 1 + 3/2 = 0

-x^3 - 5x^2 - 3x + 1/2 = 0

The characteristic equation is: -x^3 - 5x^2 - 3x + 1/2 = 0

To find the eigenvalues, we need to solve the characteristic equation. This can be done by graphing or by using numerical methods such as the Newton-Raphson method or the bisection method.