Problem 31 - 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 \(A .\) ) $$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

The polynomial is \( x^2 - 3x - 4 \), and the zeros (eigenvalues) are 4 and -1.

Step 1: Understand the Problem

We need to find the characteristic polynomial \( f(x) \) for the given matrix \( A \) and then determine its zeros, which are the eigenvalues of \( A \). The characteristic polynomial is found by calculating the determinant \( \det(A - xI) \). Here, \( I \) is the identity matrix of the same order as \( A \).

Step 2: Write the Identity Matrix

Since \( A \) is a 2x2 matrix, the identity matrix \( I_2 \) is:\[I_2 = \begin{bmatrix} 1 & 0 \0 & 1 \end{bmatrix}\]

Step 3: Formulate the Matrix \( A-xI \)

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

Step 4: Calculate the Determinant \( \det(A-xI) \)

Use the formula for the determinant of a 2x2 matrix, \( \det \begin{bmatrix} a & b \c & d \end{bmatrix} = ad - bc \):\[\det(A - xI) = (1-x)(2-x) - (3)(2)\]\[= (1-x)(2-x) - 6\]

Step 5: Expand the Expression

Expand the expression from the determinant calculation:\[(1-x)(2-x) = 2 - 3x + x^2\]Thus, the polynomial becomes:\[\det(A - xI) = x^2 - 3x + 2 - 6 = x^2 - 3x - 4\]

Step 6: Find the Zeros of the Polynomial

Solve \( f(x) = x^2 - 3x - 4 = 0 \) for \( x \). Factor the quadratic equation:\[x^2 - 3x - 4 = (x-4)(x+1) = 0\]Thus, \( x = 4 \) and \( x = -1 \) are the zeros of the polynomial.

Key Concepts

Eigenvalues

Eigenvalues are a fundamental concept in linear algebra, used to analyze matrices. When we talk about eigenvalues, we refer to the characteristic values that can "scale" a vector when multiplied by a particular matrix. For a given matrix \( A \), its eigenvalues are found by solving the characteristic polynomial, which comes from the expression \( \det(A - xI) = 0 \). These values are where the determinant of the difference between the matrix \( A \) and \( x \) times the identity matrix is zero.

Eigenvalues provide insight into the properties of linear transformations, and they are particularly useful in simplifying complex matrix equations. In many cases, they reveal intrinsic information about the matrix, like stability in systems or modes of vibration in structures.

To find the eigenvalues, follow these general steps:

• Write the identity matrix \( I \) of the same order as matrix \( A \).
• Subtract \( xI \) from \( A \) to form \( A-xI \).
• Calculate the determinant of \( A-xI \).
• Solve the resulting characteristic equation for \( x \).

Determinant

The determinant of a matrix is a special number that can provide lots of information about the matrix itself. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). This mathematical expression gives us a scalar (a single number) that tells us various properties of the matrix, such as singularity and invertibility.

In the context of finding eigenvalues, the determinant of matrices like \( A-xI \) is essential because:

• When the determinant is zero, the matrix \( A \) does not have an inverse.
• The points where \( \det(A-xI) = 0 \) are precisely the eigenvalues of \( A \).

For the matrix \( A \) in our example, the determinant of \( A-xI \) leads us to the polynomial \( x^2 - 3x - 4 \), whose zeros give the eigenvalues \( x = 4 \) and \( x = -1 \). This step is crucial in solving for eigenvalues, as it provides the roots we need.

Identity Matrix

The identity matrix, often denoted by \( I \), is a vital concept in matrix algebra. It plays a role similar to the number 1 in multiplication, serving as a neutral element. For any matrix \( A \) and identity matrix \( I \) of the same size, multiplying \( A \) by \( I \) results in \( A \) itself, i.e., \( IA = AI = A \). In a 2x2 matrix, the identity matrix is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

When finding the characteristic polynomial, \( xI \) is subtracted from the matrix \( A \) to create \( A-xI \). This step is essential for determining the matrix's eigenvalues. It translates the concept of "eigen" (or inherent) values from the realm of transformations into the mathematically tractable form of polynomial roots. The utility of the identity matrix lies in its simplicity and its ability to make complex calculations more manageable by providing a clear basis for matrix transformations.

Matrix Subtraction

Matrix subtraction is a straightforward operation where corresponding elements of two matrices are subtracted. It follows basic arithmetic rules with one key condition: both matrices must be of the same order. When subtracting two matrices, say \( A \) and \( B \), each entry \( a_{ij} \) from matrix \( A \) is paired with entry \( b_{ij} \) from matrix \( B \), resulting in a new matrix \( C \) where each \( c_{ij} = a_{ij} - b_{ij} \).

In our example, matrix subtraction is applied to compute \( A-xI \), which is crucial in finding the characteristic polynomial. Here's how it is structured:

• Consider the original matrix \( A \).
• Form \( xI \) by scaling the identity matrix \( I \) by \( x \).
• Subtract \( xI \) from \( A \) to get \( A-xI \).

This operation modifies each element of the original matrix according to \( x \), setting the stage for determinant calculation. The resulting matrix form is directly used to find where \( \det(A-xI) \) becomes zero, revealing the eigenvalues of the matrix.