Problem 61 - Calculus for Biology and Medicine

Question

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & 0 \\ c & b \end{array}\right] $$

Step-by-Step Solution

Answer

The eigenvalues are \( \lambda_1 = a \) and \( \lambda_2 = b \).

Step 1: Write the Characteristic Equation

The characteristic equation is obtained by subtracting \( \lambda I \) from the matrix \( A \) and setting the determinant to zero. Here, \( I \) is the identity matrix. For a 2x2 matrix \( A = \begin{bmatrix} a & 0 \ c & b \end{bmatrix} \), the equation is:\[ |A - \lambda I| = |\begin{bmatrix} a - \lambda & 0 \ c & b - \lambda \end{bmatrix}| = 0 \]

Step 2: Calculate the Determinant

Expand the determinant for the matrix:\[ |\begin{bmatrix} a - \lambda & 0 \ c & b - \lambda \end{bmatrix}| = (a - \lambda)(b - \lambda) - (0)(c) = (a - \lambda)(b - \lambda) \]

Step 3: Solve the Characteristic Equation

The characteristic equation \((a - \lambda)(b - \lambda) = 0\) implies that either \( (a - \lambda) = 0 \) or \( (b - \lambda) = 0 \). Solve for \( \lambda \):- \( a - \lambda = 0 \) gives \( \lambda = a \).- \( b - \lambda = 0 \) gives \( \lambda = b \).

Step 4: Identify the Eigenvalues

The solutions from the characteristic equation, \( \lambda = a \) and \( \lambda = b \), are the eigenvalues of the matrix \( A \). Therefore, the eigenvalues \( \lambda_1 \) and \( \lambda_2 \) are \( \lambda_1 = a \) and \( \lambda_2 = b \).

Key Concepts

Characteristic Equation

When dealing with matrices, a very important concept is the characteristic equation. This equation is crucial in finding the eigenvalues of a matrix, which are fundamental in various fields of mathematics and engineering. The characteristic equation is derived from a square matrix by subtracting a scalar multiple of the identity matrix from the matrix itself and then calculating the determinant. For a 2x2 matrix, the identity matrix is \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\).
To formulate the characteristic equation, you replace the diagonal elements of the matrix with \( (a - \lambda) \) and \( (b - \lambda) \) where \( \lambda \) is an unknown scalar, representing the eigenvalue. You then equate the determinant of this new matrix to zero. Solving this equation allows you to find the eigenvalues of the original matrix effectively.
Understanding the characteristic equation can greatly assist in comprehending the behavior of matrices and their applications in transformations, stability analysis, and more.

2x2 Matrix

A 2x2 matrix is a fundamental building block in linear algebra. It consists of two rows and two columns, represented generally as: \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\).
These matrices are some of the simplest yet powerful concepts as they provide a glimpse into more complex structures. Operations such as addition, multiplication, and finding inverses, all occur in the same basic way, allowing these matrices to be a great learning tool.

• They represent linear transformations in a two-dimensional space.
• They can model systems with two variables like coupled equations.
• They offer insights into broader concepts such as trace, determinant, and eigenvalues.

2x2 matrices allow you to perform calculations that can describe rotations, dilations, and reflections. Despite their simplicity, they underpin many real-world applications across physics, computer graphics, and statistics.

Determinant

The determinant of a matrix provides critical information about a matrix. Specifically for a 2x2 matrix, the determinant can give insights into the matrix's invertibility and the solutions of the system of equations they model. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) can be calculated using the formula: \[\text{det}(A) = ad - bc\].
In our specific case of finding eigenvalues, the determinant helps solve the characteristic equation. For example, when considering the matrix \(A = \begin{bmatrix} a & 0 \ c & b \end{bmatrix}\),
taking the determinant as part of finding the eigenvalues simplifies to comparing products of diagonal elements. Understanding determinants helps establish conditions like:

• If the determinant is zero, the matrix is singular and does not have an inverse.
• If non-zero, the matrix is non-singular and invertible.

This property is vital for systems' stability and behavior, indicating how solutions react within mathematical, physical, or engineering frameworks.