Problem 33 - 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 \(A .\) ) $$A=\left[\begin{array}{rr} -3 & -2 \\ 2 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x^2 + x - 2\), and its zeros are \(-2\) and \(1\).

Step 1: Define the Characteristic Polynomial

The identity matrix \(I_2\) for a \(2\times2\) matrix is \(\begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\). To find \(f(x)\), compute \(|{\mathbf{A} - x\mathbf{I}}|\), where \(\mathbf{A}\) is the given matrix and \(x\) is a scalar. Substitute \(\mathbf{A}\) and \(\mathbf{I}\) into the expression: \[ \mathbf{A} - x\mathbf{I} = \begin{bmatrix} -3 - x & -2 \ 2 & 2 - x \end{bmatrix} \].

Step 2: Find the Determinant

Compute the determinant of the matrix \(\mathbf{A} - x\mathbf{I}\): \[|\mathbf{A} - x\mathbf{I}| = (-3-x)(2-x) - (-2)(2)\]Calculating, we get: \[= (-3-x)(2-x) + 4\]

Step 3: Expand the Expression

Expand the expression obtained from the determinant: \[=(-3)(2-x) - x(2-x) + 4\]Simplifying further, we compute:\[= (-6 + 3x) - (2x - x^2) + 4\]\[= -6 + 3x - 2x + x^2 + 4\]

Step 4: Combine Like Terms

Combine all like terms from the expanded expression to form the polynomial: \[= x^2 + x - 2\] Thus, the characteristic polynomial is \(f(x) = x^2 + x - 2\).

Step 5: Find the Zeros of the Polynomial

To find the zeros of \(f(x)\), solve the equation \(x^2 + x - 2 = 0\). This can be factored into:\[(x+2)(x-1) = 0\]

Step 6: Solve for Roots

Set each factor equal to zero:1. \(x+2 = 0\) gives \(x = -2\)2. \(x-1 = 0\) gives \(x = 1\)Therefore, the solutions or zeros are \(-2\) and \(1\).

Key Concepts

Determinant

The notion of the determinant is a vital concept in linear algebra, especially when working with matrices. In simple terms, the determinant of a square matrix is a unique number that can be computed from its elements. It is denoted as \(|\mathbf{A}|\) for a matrix \(\mathbf{A}\). For a 2x2 matrix, the determinant is calculated as follows: if \(\mathbf{A} = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is \(ad - bc\). This determinant helps in determining if a matrix is invertible; a matrix is invertible if its determinant is not zero.
In the context of finding the characteristic polynomial, we compute the determinant of the matrix \(\mathbf{A} - x\mathbf{I}\), where \(x\) is a scalar, and \(\mathbf{I}\) is the identity matrix. The result, a polynomial, is what we call the characteristic polynomial of \(\mathbf{A}\).
Determinants prove useful beyond calculating invertibility; they play a crucial role in finding eigenvalues, solving systems of linear equations, and working with transformations in vector spaces.

Eigenvalues

Eigenvalues are essential in understanding the behavior of linear transformations. They are special scalar values that provide insight into the properties of a matrix. When we talk about eigenvalues, we're dealing with values of \(x\) that satisfy the equation \(|\mathbf{A} - x\mathbf{I}| = 0\). These values are the roots of the characteristic polynomial.
In simple terms, if \(\mathbf{A}\) is a square matrix and \(\mathbf{v}\) is a non-zero vector, there exists a scalar \(\lambda\) (the eigenvalue) such that \(\mathbf{A}\mathbf{v} = \lambda\mathbf{v}\). In essence, eigenvalues tell us how a transformation defined by \(\mathbf{A}\) stretches or shrinks vectors in its vector space. They help in understanding whether a matrix is stable, and they are crucial in practical applications like vibration analysis and image processing.

Matrix Algebra

Matrix algebra forms the backbone of many computations in linear algebra. Matrices are organized into rows and columns, representing systems of linear equations or transformations of vector spaces. Basic operations such as addition, subtraction, and multiplication, including with scalars, shape how we use matrices in applications.
One key operation is multiplying matrices. For two matrices \(\mathbf{A}\) and \(\mathbf{B}\), the element in the \((i, j)\) position of \(\mathbf{AB}\) is the dot product of the \(i\)th row of \(\mathbf{A}\) and the \(j\)th column of \(\mathbf{B}\).

• Matrix addition is straightforward by adding corresponding elements of the matrices.
• Scalar multiplication involves multiplying every element by the scalar value.
• Multiplying a matrix by the identity matrix does not change the original matrix.

Understanding these operations is vital for applications in various fields, including computer graphics, economics, and quantum mechanics.

Identity Matrix

The identity matrix is an important element in matrix algebra. It operates as a neutral element in matrix multiplication. For a matrix of size \(n\times n\), the identity matrix \(\mathbf{I}_n\) is a matrix with ones on its diagonal and zeros elsewhere. Mathematically, \(\mathbf{I}_n = \begin{bmatrix} 1 & 0 & 0 & \cdots \ 0 & 1 & 0 & \cdots \ \vdots & \ddots & \ddots & \vdots \end{bmatrix}\).
Multiplying any matrix \(\mathbf{A}\) by its corresponding identity matrix dimensions results in the original matrix \(\mathbf{A}\). For example, \(\mathbf{AI} = \mathbf{A}\) and \(\mathbf{IA} = \mathbf{A}\). This special property makes the identity matrix pivotal, particularly in definitions like the characteristic polynomial, where \(\mathbf{A} - \mathbf{xI}\) involves scalar multiplication of a matrix by the identity matrix.
The role of the identity matrix in transformation exercises and equivalence of matrices cannot be overstated, as it essentially defines the inherent structure and consistency sought in matrix solutions.