Problem 48 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The polynomial is \(f(x) = -x^3 + 2x^2 + 5x - 6\); the characteristic values are approximately \(-3, 1, 2.\)

Step 1: Define the Identity Matrix

For a matrix of order 3, the identity matrix \(I\) is given by \(I_{3} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\).

Step 2: Form the Matrix \(A - xI\)

Subtract \(x\) times the identity matrix from \(A\): \[ A - xI = \begin{bmatrix} 3-x & -1 & -1 \ -1 & 1-x & 0 \ -1 & 0 & -2-x \end{bmatrix} \]

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

To find \(f(x)\), compute the determinant of \(A - xI\). Use the cofactor expansion: \[|A - xI| = (3-x)((1-x)(-2-x) - (0 \cdot 0)) - (-1)((-1)(-2-x) - (0 \cdot -1)) + (-1)((-1 \cdot 0) - (1-x)(-1))\]

Step 4: Simplify the Determinant Expression

Carrying out the expansion: 1. Expand \((1-x)(-2-x) = (1-x)(-2-x) = -2 - x + 2x + x^2 = x^2 + x - 2\).2. The term involving \(-1\) becomes \((-1)(-2-x) = 2 + x\).3. The third part becomes \(1-x\).Thus, the determinant simplifies to: \[ (3-x)(x^2 + x - 2) + (2 + x) - (1-x) \]

Step 5: Expand and Simplify the Polynomial

Expand the polynomial: 1. \((3-x)(x^2 + x - 2) = 3(x^2 + x - 2) - x(x^2 + x - 2)\)2. Expand both terms: - \(3x^2 + 3x - 6\) - \(-x^3 - x^2 + 2x\)3. Combine all terms: \[-x^3 + 2x^2 + 5x - 6\]. Thus, \(f(x) = -x^3 + 2x^2 + 5x - 6\).

Step 6: Graph the Function \(f(x)\)

Plotting the polynomial \(f(x) = -x^3 + 2x^2 + 5x - 6\) shows a cubic curve. The roots (characteristic values) are approximately the values of \(x\) where \(f(x)\) crosses the x-axis.

Step 7: Estimate Roots for Characteristic Values

Estimate the root values by looking at the graph. They correspond to points where \(f(x) = 0\). The roots look around \(x \approx -3, 1,\) and \(2\). These are the approximate eigenvalues or characteristic values of \(A\).

Key Concepts

Eigenvalues

Eigenvalues are a fundamental concept in linear algebra, essential for understanding the properties of matrices. In the context of the given exercise, eigenvalues are the characteristic values of matrix \(A\), where the characteristic polynomial \(f(x)\) is evaluated to zero.

• To find eigenvalues, we calculate \(|A - xI| = 0\), where \(I\) is the identity matrix and \(x\) represents the eigenvalue.
• The roots of this equation are the eigenvalues of the matrix \(A\).

The significance of eigenvalues is vast; they can indicate stability in systems, allow matrix diagonalization, and are used in various applications such as vibrations analysis, control systems, and even Google's PageRank algorithm. Grasping how they are calculated and interpreted gives us powerful insights into the matrix's behavior.

Matrix Determinant

The determinant of a matrix is a unique number that can be calculated from its square matrix. It provides vital information about the matrix, such as invertibility and volume scaling factor.

• In solving for the characteristic polynomial \(f(x)\), the determinant \(|A - xI|\) is computed.
• The condition \(|A - xI| = 0\) is set for finding the eigenvalues, reflecting that the matrix loses invertibility at these points.

Calculating determinants can be tricky, especially for larger matrices, but it's essential for understanding their deeper properties. In practical applications, determinants can also be used in solving systems of linear equations using Cramer's Rule. They also play a crucial role in changing variables in multiple integrals - showing their utility beyond just pure matrices.

Cofactor Expansion

Cofactor Expansion is a method used to calculate the determinant of a matrix. It involves expanding the determinant along a row or column, using minors and cofactors.

• For a 3x3 matrix, choosing any row or column allows us to break down the determinant into simpler smaller parts.
• The cofactor expansion employed in the exercise expands along the first row, leading to simpler calculations for \(|A - xI|\).

This method is not only applicable but essential for grasping how determinants work for larger matrices. It simplifies calculations into manageable pieces, and understanding it can be a game-changer in applied mathematics, where large datasets can cause computational strain.

Cubic Functions

Cubic functions are polynomial equations of degree three, and they exhibit unique properties due to their highest exponent being three. In this exercise, \(f(x) = -x^3 + 2x^2 + 5x - 6\) is a cubic function and represents the characteristic polynomial of the matrix \(A\).

• These functions often have up to three real roots, as highlighted in the roots \(x \approx -3, 1, 2\).
• The graph of a cubic function is a curve that can show different inflection points, crucial for deducing the behavior and intersection with the axes.

Understanding cubic functions and their graphs is instrumental in interpreting the solution to our problem. From physics in understanding movement through a curve to economics in modeling cost functions, cubic functions are a gateway to diverse analytical fields.