Problem 34 - 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 \(A .\) ) $$A=\left[\begin{array}{rr} 2 & -4 \\ -3 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

The characteristic polynomial is \(x^2 - 7x - 2\) and its zeros are \(x = \frac{7 \pm \sqrt{57}}{2}.\)

Step 1: Identify the Identity Matrix

The identity matrix \( I_2 \) of order 2 is given by \(I = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right].\)

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

Subtract \(xI\) from \(A\):\[A - xI = \left[\begin{array}{cc} 2 & -4 \ -3 & 5 \end{array}\right] - \left[\begin{array}{cc} x & 0 \ 0 & x \end{array}\right] = \left[\begin{array}{cc} 2-x & -4 \ -3 & 5-x \end{array}\right].\]

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

Calculate the determinant of the matrix \(A - xI\):\[|A - xI| = (2-x)(5-x) - (-4)(-3).\]Multiply the components to find:\[(2-x)(5-x) = (10 - 7x + x^2)\quad \text{and}\quad (-4)(-3) = 12.\]Thus,\[|A-xI| = x^2 - 7x - 2.\]

Step 4: Establish the Characteristic Polynomial

From the determinant, we know that the characteristic polynomial is:\[f(x) = x^2 - 7x - 2.\]

Step 5: Find the Zeros/Roots of the Polynomial

To find the zeros of the polynomial \(f(x)\), solve the equation\[x^2 - 7x - 2 = 0.\]Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-7\), \(c=-2\):\[x = \frac{7 \pm \sqrt{(-7)^2 - 4 \times 1 \times (-2)}}{2 \times 1} = \frac{7 \pm \sqrt{49 + 8}}{2} = \frac{7 \pm \sqrt{57}}{2}.\]So the solutions are \(x = \frac{7 + \sqrt{57}}{2}\) and \(x = \frac{7 - \sqrt{57}}{2}.\)

Key Concepts

Identity Matrix

The identity matrix is a fundamental concept in linear algebra. It is often represented by the letter \( I \). In its essence, the identity matrix acts like the number 1 does in multiplication. For a square matrix of order \( n \), the identity matrix is \( n \times n \) and contains 1s along the main diagonal (from top left to bottom right), and 0s elsewhere.

• For a 2x2 identity matrix \( I_2 \), it looks like:\[I_2 = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]
• It doesn't change a vector when multiplied by it. If you multiply any matrix \( A \) by \( I \), you will get \( A \) back. This is expressed as \( A \cdot I = I \cdot A = A \).

Understanding the identity matrix helps in determining characteristic polynomials and finding eigenvalues.

Determinant

The determinant is a scalar value that can be calculated from a square matrix. It provides essential insights into the properties of the matrix, such as its invertibility and eigenvalues.

• For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as:\[|A| = ad - bc\]
• The determinant can tell us if a matrix is invertible. If the determinant is 0, the matrix does not have an inverse.
• For eigenvalue problems, the determinant of \( A - xI \) (where \( A \) is a matrix and \( x \) is a scalar) provides the characteristic polynomial \( f(x) \).

In the exercise, calculating the determinant of \( A - xI \) is a crucial step in forming the characteristic polynomial.

Eigenvalues

An eigenvalue is a special scalar associated with a linear transformation represented by a matrix. These values are critical in understanding the matrix's inherent characteristics.

• The eigenvalues of a matrix \( A \) are the solutions to the equation \( f(x) = 0 \), where \( f(x) \) is the characteristic polynomial.
• In the exercise, after calculating the determinant to obtain the characteristic polynomial \( f(x) = x^2 - 7x - 2 \), solving this polynomial gives the eigenvalues.
• Eigenvalues might give you insight into the behavior of physical systems, stability of equilibrium points, and other applications in various fields.

In short, eigenvalues are key to gaining deeper insights into matrix behaviors.

Quadratic Formula

The quadratic formula is a mathematical tool that provides the solutions (roots) of a quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is very handy for finding eigenvalues when the characteristic polynomial is quadratic.

• The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• In the context of the exercise, we used the quadratic formula to solve \( x^2 - 7x - 2 = 0 \), which resulted from the determinant calculation.
• The solutions \( x = \frac{7 \pm \sqrt{57}}{2} \) are the eigenvalues of matrix \( A \).

This formula is so versatile, providing a straightforward way to find solutions for any quadratic equation, which is crucial in eigenvalue problems.