Question

Find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector. $$\left[\begin{array}{ccc}1 & 0 & -18 \\ -4 & 3 & -1 \\ 1 & 0 & -8\end{array}\right]$$

Short answer

The eigenvalues are 3, 2, -9 with corresponding eigenvectors \([9, 1, 1]\), \([9, 1, 1]\), \([2, 1, 1]\) respectively.

Step-by-step solution

Find the Characteristic Polynomial

The first step is to find the characteristic polynomial of the matrix. The characteristic polynomial is found by evaluating \( \det(A - \lambda I) \), where \( A \) is the matrix in question and \( I \) is the identity matrix.\( A = \begin{bmatrix} 1 & 0 & -18 \ -4 & 3 & -1 \ 1 & 0 & -8 \end{bmatrix}, \ I = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). Substituting into the formula, \( A - \lambda I = \begin{bmatrix} 1-\lambda & 0 & -18 \ -4 & 3-\lambda & -1 \ 1 & 0 & -8-\lambda \end{bmatrix} \).\ Therefore, the determinant is \[ \det \begin{bmatrix} 1-\lambda & 0 & -18 \ -4 & 3-\lambda & -1 \ 1 & 0 & -8-\lambda \end{bmatrix} = (1-\lambda)((3-\lambda)(-8-\lambda) - (0)(-1)) - (0)((-4)(-8-\lambda) - (-1)(1)) + (-18)((-4)(0) - (3-\lambda)(1)) \].

Compute the Determinant

Solving the determinant expression, \( = (1-\lambda)((3-\lambda)(-8-\lambda)) - 0 + 18(3-\lambda) \).\ Simplifying: \( = (1-\lambda)((\lambda^2 + 5\lambda - 24)) + 54 - 18\lambda \).\ Further simplifying gives the characteristic polynomial \( \lambda^3 + 5\lambda^2 - (24+18)\lambda + 54 = \lambda^3 + 5\lambda^2 - 42\lambda + 54 \).

Solve the Characteristic Polynomial for Eigenvalues

The characteristic equation is \( \lambda^3 + 5\lambda^2 - 42\lambda + 54 = 0 \). Solving this cubic polynomial can be done using methods such as factoring, the Rational Root Theorem, or through numerical approximation or graphing.\ After solving, the eigenvalues are found to be \( \lambda_1 = 3, \lambda_2 = 2, \lambda_3 = -9 \).

Find Eigenvectors

For each eigenvalue, we substitute it back into \( A - \lambda I \) and solve \((A - \lambda I)X = 0 \) to find the eigenvectors.\ For \( \lambda_1 = 3 \): \( \begin{bmatrix} -2 & 0 & -18 \ -4 & 0 & -1 \ 1 & 0 & -11 \end{bmatrix}X = 0 \), leading to eigenvector \( \begin{bmatrix} 9 \ 1 \ 1 \end{bmatrix} \).\ For \( \lambda_2 = 2 \): \( \begin{bmatrix} -1 & 0 & -18 \ -4 & 1 & -1 \ 1 & 0 & -10 \end{bmatrix}X = 0 \), leading to eigenvector \(\begin{bmatrix} 9 \ 1 \ 1 \end{bmatrix} \). \ For \( \lambda_3 = -9 \): \( \begin{bmatrix} 10 & 0 & -18 \ -4 & 12 & -1 \ 1 & 0 & 1 \end{bmatrix}X = 0 \), which simplifies to eigenvector \( \begin{bmatrix} 2 \ 1 \ 1 \end{bmatrix} \).

Key concepts

Characteristic Polynomial

The characteristic polynomial is an essential tool when finding the eigenvalues of a matrix. It relates a square matrix to its eigenvalues, which are intrinsically linked to the transformations that the matrix represents. By definition, the characteristic polynomial of a matrix \( A \) is obtained by calculating the determinant of \( A - \lambda I \), where \( \lambda \) is a scalar, and \( I \) is the identity matrix. This calculation results in a polynomial expression dependent on the variable \( \lambda \).

• The polynomial is obtained by substituting \( \lambda \) into the determinant equation \( \det(A - \lambda I) \).
• Its degree is equivalent to the size of the matrix; for a 3x3 matrix, the characteristic polynomial is cubic.
• Finding the zeros of this polynomial gives the eigenvalues of the matrix.

Understanding and finding the characteristic polynomial provide the foundation for determining eigenvalues, which reflect how vectors stretch or shrink under the linear transformation dictated by the matrix. It's crucial in various applications including stability in systems and vibrations in mechanical structures.

Determinant Calculation

Determinant calculation plays a vital role in obtaining the characteristic polynomial of a matrix. It's a mathematical expression of a matrix used to evaluate various properties, such as invertibility or the volume transformation carried out by the matrix. In the context of eigenvalue problems, the determinant is specifically used to identify eigenvalues by setting \( \det(A - \lambda I) = 0 \).

• For a 3x3 matrix, finding the determinant involves expanding along a row or column and calculating the associated minors.
• The determinant of \( A - \lambda I \) is simplified using the properties of determinants, such as linearity and multi-linearity.
• Each step of simplification narrows down the polynomial, leading to the characteristic polynomial.

Once the determinant is calculated, it provides a polynomial that serves as the groundwork for finding eigenvalues by seeking the roots of this polynomial expression. The calculated determinant not only is pivotal in defining eigenvalues but also complements understanding the behavioral aspects of the matrix.

Cubic Polynomial Solutions

Solving a cubic polynomial is essentially finding the roots of a third-degree polynomial, which correspond to the eigenvalues of a matrix when dealing with its characteristic polynomial. Solving these polynomials is crucial in numerous scientific and engineering problems where understanding the behavior of systems is essential.

• Cubic polynomials take the form \( \lambda^3 + a_2\lambda^2 + a_1\lambda +a_0 = 0 \).
• There are several methods available, including factoring, the Rational Root Theorem, and numerical solutions.
• Without complex equations involved, graphing the polynomial can also offer visual solutions through intersection points.

Each method has unique strengths—factoring is efficient with simple roots, while approximate methods are suited for complex or non-factorable roots. Successfully solving the cubic polynomial provides the exact eigenvalues necessary to define the behavior of a matrix in various physical and mathematical applications. Understanding these solutions helps reveal how different vectors are transformed, offering insight into the underlying characteristics of the system modeled by the matrix.