Problem 36 - 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 \(f(x)\) $$A=\left[\begin{array}{rrr} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 1 & 3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

(a) The polynomial is \(-x^3 + 4x^2 + 4x - 2\). (b) Zeros can be found using numerical methods.

Step 1: Define the Matrix I and f(x)

We are given that \(I = I_3\), the 3x3 identity matrix, and we define the function \(f(x) = |A - xI|\). Here, \(A\) is the given matrix and \(x\) is a scalar.

Step 2: Construct the Matrix (A - xI)

Substitute \(I\) with the 3x3 identity matrix in the expression \(A - xI\) to get:\[A - xI = \begin{bmatrix} 2 & 1 & 0 \ -1 & 0 & 0 \ 1 & 3 & 2 \end{bmatrix} - \begin{bmatrix} x & 0 & 0 \ 0 & x & 0 \ 0 & 0 & x \end{bmatrix} = \begin{bmatrix} 2-x & 1 & 0 \ -1 & -x & 0 \ 1 & 3 & 2-x \end{bmatrix}\]

Step 3: Find the Determinant |A - xI|

Calculate the determinant \(|A-xI|\) using the matrix obtained:\[\left| \begin{array}{ccc} 2-x & 1 & 0 \ -1 & -x & 0 \ 1 & 3 & 2-x \end{array} \right| = (2-x)(-x(2-x)) - 1(-(2-x)) - 0 = (2-x)(x^2 - 2x) + (2-x)\]Expand this to get the polynomial expression.

Step 4: Expand and Simplify the Polynomial

Simplify the expression for the determinant:\[(2-x)x^2 - 2(2-x)x + (2-x) = (2-x)x^2 - 4x + 2 = -x^3 + 4x^2 + 4x - 2\]This is the polynomial \(f(x)\).

Step 5: Find the Zeros of f(x)

To find the zeros of \(f(x) = -x^3 + 4x^2 + 4x - 2\), solve the equation \(f(x) = 0\) using methods such as factoring, synthetic division, or numerical solutions. The zeros can be approximate or exact, as appropriate.

Step 6: Factor and Solve the Polynomial

Attempt to factor the polynomial or use a method like the Rational Root Theorem to find possible roots. Since factoring does not readily yield simple results, we may use numerical methods or graphing techniques.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is an important concept in matrix algebra, representing a scalar value associated with a square matrix. Understanding the determinant can aid in solving systems of linear equations, finding eigenvalues, and determining invertibility of a matrix.

To calculate the determinant, particularly for a 3x3 matrix, follow a process called cofactor expansion. Utilize the formula:\[|A| = a(ei-fh) - b(di-fg) + c(dh-eg)\]where \(a, b, c\) are elements of the first row, and each pair such as \(ei, fh\) corresponds to another set of elements forming a 2x2 matrix from the remaining rows and columns.

In our example, transforming matrix \((A-xI)\), we calculate:\[(2-x)(x^2 - 2x) + (2-x)\]to find the polynomial determinant that helps us later find the characteristic polynomial used for determining eigenvalues.

Zeros of a Polynomial

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. Finding these values, also known as roots, is crucial in understanding the behavior and solutions of polynomial equations.

For the polynomial \(f(x) = -x^3 + 4x^2 + 4x - 2\), set it equal to zero:\[-x^3 + 4x^2 + 4x - 2 = 0\]To discover the zeros, techniques such as:

• Factoring, to express the polynomial in a product of its simplest expressions
• Using Rational Root Theorem, which suggests possible rational roots based on the factors of the constant and leading coefficient
• Graphing, to visually estimate approximate roots
• Numerical methods, for solving complex polynomials where zeros aren't easy to find manually

These approaches may yield either exact or approximate roots, depending on the complexity of the polynomial, thus providing valuable solutions.

Identity Matrix

An identity matrix is a special kind of square matrix that serves as the multiplicative identity in matrix algebra, much like the number 1 does in arithmetic. It is denoted as \(I_n\) for an \(n \times n\) matrix.

An identity matrix has ones on the diagonal from the top left to bottom right, and zeros elsewhere. A 3x3 identity matrix looks like this:\[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\]
The key property of the identity matrix is that when any matrix \(A\) is multiplied by \(I\), the result is the original matrix \(A\) itself:\[AI = IA = A\]In the exercise, \(I_3\) is utilized to construct the matrix \((A-xI)\), which is essential for forming the characteristic polynomial when determining eigenvalues.

Matrix Algebra

Matrix algebra involves operations on matrices, crucial for solving linear equations and modeling various systems. The foundation includes operations such as addition, subtraction, multiplication, and finding inverses and determinants.

• Addition/Subtraction: Combine matrices by adding or subtracting corresponding elements.
• Multiplication: Multiply by considering the dot product of rows and columns. The result of an \(m \times n\) and an \(n \times p\) matrix multiplication yields an \(m \times p\) matrix.
• Inversion: Find the matrix inverse such that when multiplied by the original matrix results in the identity matrix.

The exercise uses subtraction in \((A - xI)\), showcasing how matrices transform when combined with scalars and other matrices. This transformation reveals important properties like determinants and characteristic polynomials, serving not only in theoretical aspects but practical applications as well.