Problem 35 - Precalculus: Functions and Graphs

Question

Let \(I=I_{3}\) and let \(f(x)=|A-x I| .\) Find (a) the polynomial \(f(x)\) and (b) the zeros of \(f(x)\) $$A=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 1 & 0 & -2 \\ -1 & 1 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) \( f(x) = -x^3 - 3x^2 + 5x - 2 \). (b) Zeros of \( f(x)\) are the roots of this polynomial.

Step 1: Understanding the Problem

We are given the matrix \( A \) and required to find the polynomial \( f(x) \) and its zeros. The function \( f(x) = |A - xI| \) represents the characteristic polynomial of the matrix \( A \). This polynomial helps us find the eigenvalues, which are the zeros of \( f(x) \).

Step 2: Set Up the Matrix \( A - xI \)

Given matrix \( A \), the identity matrix \( I_3 \) is \( \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \). Subtract \( xI \) from \( A \) to find \( A - xI \): \( A - xI = \begin{bmatrix} 1-x & 0 & 0 \ 1 & 0-x & -2 \ -1 & 1 & -3-x \end{bmatrix} \).

Step 3: Find the Determinant of \( A - xI \)

The characteristic polynomial \( f(x) \) is obtained by computing the determinant \( |A - xI| \). This involves calculating: \[|A - xI| = \left| \begin{array}{ccc} 1-x & 0 & 0 \ 1 & -x & -2 \ -1 & 1 & -3-x \end{array} \right|.\]Evaluate this determinant using cofactor expansion along the first row.

Step 4: Compute Cofactor Expansion

Perform the cofactor expansion along the first row:\[|A - xI| = (1-x) \left| \begin{array}{cc} -x & -2 \ 1 & -3-x \end{array} \right| = (1-x)(x(-3-x) - (-2))\]which simplifies to \[(1-x)(-x^2 - 3x + 2).\]

Step 5: Simplify the Polynomial

Simplify the expression from Step 4: \[f(x) = (1-x)(-x^2 - 3x + 2) = -x^3 - 3x^2 + 2x + x^3 + 3x - 2.\]This simplifies further to:\[f(x) = -x^3 + x^3 - 3x^2 + 3x^2 + 2x + 3x - 2 = -x^3 - 3x^2 + 5x - 2.\]

Step 6: Solve for the Zeros of \( f(x) \)

The zeros of \( f(x) = -x^3 - 3x^2 + 5x - 2 \) are the eigenvalues of \( A \). Solve \( f(x) = 0 \) by factoring or using numerical methods if necessary. The roots will generally need a calculator or solving software to approximate.

Key Concepts

Understanding Eigenvalues

Eigenvalues are a fundamental concept in linear algebra, often appearing when dealing with matrices. These are special scalars associated with a matrix that provide important insights into its properties, such as stability and dynamics. If we have a square matrix \(A\) and a non-zero vector \(\mathbf{v}\), an eigenvalue \(\lambda\) satisfies the equation \(A \mathbf{v} = \lambda \mathbf{v}\). This equation tells us that applying the matrix \(A\) to the vector \(\mathbf{v}\) has the same effect as scaling \(\mathbf{v}\) by some factor \(\lambda\).

• Eigenvalues help to understand the behavior of a system represented by a matrix.
• In a characteristic polynomial, the roots are the eigenvalues.
• Finding eigenvalues is crucial for solving equations involving matrices, such as systems of linear equations.

The zeros of the characteristic polynomial give us the eigenvalues, essentially telling us how the matrix transformation behaves.

Determinant and Its Role

The determinant of a matrix is a special number that can provide a lot of information about the matrix. For any square matrix, the determinant is a scalar value that is computed from its elements and is denoted as \(|A|\). It's a crucial tool in matrix algebra to determine when a matrix is invertible, as a matrix is invertible if and only if its determinant is non-zero.

In the context of finding eigenvalues, the determinant of \(A - xI\), where \(I\) is the identity matrix and \(x\) is a scalar, becomes particularly important. The characteristic polynomial is formed by calculating this determinant, \(|A - xI|\), and finding its roots corresponds to finding the eigenvalues.

• The determinant is used to simplify systems of equations.
• It is a way to verify if a matrix has an inverse.

By setting \(|A - xI| = 0\), we derive the characteristic polynomial whose roots are the eigenvalues of the matrix \(A\).

Cofactor Expansion Simplified

Cofactor expansion, also known as Laplace's expansion, is a method for calculating the determinant of a matrix. This technique involves breaking down a large determinant into smaller ones, making the calculation manageable. For a matrix \(A\),

• Choose a row or column to expand along (usually with many zeros for simplicity).
• Calculate the determinant of the minor matrices, which are the submatrices formed by removing the row and column of each element.
• Apply the formula: \(|A| = a_{11}C_{11} - a_{12}C_{12} + \ldots\)

Cofactor expansion along the first row is usually the most effective for matrices with zeros, as these simplify the process significantly. For our matrix \(A-xI\), expanding along the first row was strategic due to the presence of zeros, simplifying the calculation of the characteristic polynomial.

Basics of Matrix Algebra

Matrix algebra forms the foundation for understanding various mathematical concepts involving matrices. It's a versatile tool for solving linear equations, transforming coordinates in space, and much more. Basic operations include matrix addition, subtraction, multiplication and finding inverses.

One of the key operations shown in our problem is the subtraction of \(xI\) from \(A\), where \(I\) is the identity matrix. This operation is essential in forming \(A-xI\), crucial for deriving the characteristic polynomial.

Understanding matrix algebra:

• Matrices can represent systems of linear equations, transformations, or data sets.
• They can be manipulated in various ways to solve equations or extract information such as eigenvalues.
• Vector spaces and linear maps are all explained using matrix algebra.

Fundamentally, mastering these operations allows for a deeper understanding of both theoretical and practical applications of matrices.