Problem 32 - Precalculus: Functions and Graphs

Question

Let \(I=I_{2}\) be the identity matrix of order 2, and let \(f(x)=|\boldsymbol{A}-\boldsymbol{x} \boldsymbol{I}| .\) Find (a) the polynomial \(f(\boldsymbol{x})\) and (b) the zeros of \(f(x)\). (In the study of matrices, \(f(x)\) is the characteristic polynomial of \(A,\) and the zeros of \(f(x)\) are the characteristic values (eigenvalues) of \(A .\) ) $$A=\left[\begin{array}{ll} 3 & 1 \\ 2 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The characteristic polynomial is \(x^2 - 5x + 4\), with zeros at \(x = 1\) and \(x = 4\).

Step 1: Define Identity Matrix

The identity matrix of order 2, \( I_2 \), is a 2x2 matrix with ones on the diagonal and zeros elsewhere. It is given by: \[ I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \]

Step 2: Define Matrix \( A - xI \)

Subtract \( xI \) from \( A \). Matrix \( A \) is given as \( \begin{pmatrix} 3 & 1 \ 2 & 2 \end{pmatrix} \). Thus, \( A - xI = \begin{pmatrix} 3-x & 1 \ 2 & 2-x \end{pmatrix} \).

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

Find the determinant of the matrix \( A - xI \), which is \( \det(A - xI) = \det\begin{pmatrix} 3-x & 1 \ 2 & 2-x \end{pmatrix} \). The determinant is calculated as: \[ (3-x)(2-x) - (1)(2) \].

Step 4: Simplify the Determinant

Expand the expression \((3-x)(2-x) - 2\) to get: \[ 6 - 3x - 2x + x^2 - 2 \]. Simplify further to: \[ x^2 - 5x + 4 \]. Thus, the characteristic polynomial \( f(x) = x^2 - 5x + 4 \).

Step 5: Find the Zeros of \( f(x) \)

The zeros of the polynomial \( x^2 - 5x + 4 \) are obtained by factoring or using the quadratic formula. The polynomial factors to \( (x-1)(x-4) \), giving zeros at \( x = 1 \) and \( x = 4 \).

Key Concepts

Identity Matrix

An identity matrix is a special kind of square matrix, which has "1s" on its main diagonal and "0s" scattered everywhere else. For any square matrix of size \( n \times n \), the identity matrix is denoted as \( I_n \).
For example, the identity matrix of order 2, \( I_2 \), looks like:

• \( I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} \)

This matrix acts as the "number 1" of matrices, because whenever a matrix is multiplied by the identity matrix, the original matrix remains unchanged. In equations involving matrices, the identity matrix serves as a neutral element, much like how 1 serves as a neutral element in multiplication for numbers. It is crucial in defining terms like inverse matrices and identifying unique properties of matrices. Identity matrices are fundamental when transforming matrices in expressions, such as in the calculation for characteristic polynomials.

Eigenvalues

Eigenvalues are special numbers associated with a matrix in linear algebra. They play a critical role when studying the properties of matrices. To find the eigenvalues of a matrix \( A \), we focus on the characteristic polynomial \( f(x) = \det(A - xI) \).
This polynomial arises by taking the determinant of the matrix \( A \) subtracted by \( x \) times the identity matrix \( I \).
Once we have this polynomial, its roots, or zeros, are the eigenvalues of \( A \). They tell us about the scaling characteristics of transformations represented by the matrix \( A \).
Specifically, when a matrix acts on its eigenvectors, it simply scales them by these eigenvalues. By finding these values, we can gain insights into properties such as system stability and behavior in dynamic systems.

Matrix Determinant

The determinant is a scalar value that is computed from a square matrix, offering insights into the matrix's properties. It is represented as \( \det(A) \) when referring to matrix \( A \). Calculating the determinant involves combining the elements of a matrix in a prescribed manner.
For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is found using the formula:

• \( \det(\begin{pmatrix} a & b \ c & d \end{pmatrix}) = ad - bc \)

The determinant provides valuable information about the matrix, such as whether it is invertible; a matrix is invertible (non-singular) if its determinant is not zero. For the characteristic polynomial, calculating \( \det(A - xI) \) gives us a vital polynomial whose roots correspond to the matrix’s eigenvalues.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree 2, generally expressed in the standard form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). Quadratic polynomials arise naturally in many contexts, including the computation of characteristic polynomials for 2x2 matrices.
These polynomials are key when dealing with matrices because they encapsulate important properties of linear transformations. Solving a quadratic polynomial can be done through various methods such as:

• Factoring the expression, when possible.
• Applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

The solutions, or roots, reflect significant values related to the problem at hand. In the context of the characteristic polynomial, these roots are the eigenvalues. Understanding how to handle quadratic polynomials is essential in fully grasping matrix behaviors and their characteristics.