Problem 13 - Advanced Engineering Mathematics

Question

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} 1-\lambda & 3 \\ 2 & 2-\lambda \end{array}\right) $$

Step-by-Step Solution

Answer

\( \lambda^2 - 3\lambda - 4 \)

Step 1: Identify the Matrix

The matrix given in the problem is a 2x2 matrix: \[ \begin{pmatrix} 1-\lambda & 3 \ 2 & 2-\lambda \end{pmatrix} \]

Step 2: Apply the Determinant Formula

For a 2x2 matrix of the form \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is calculated using the formula: \( ad - bc \).

Step 3: Substitute Values

Substitute the values from the matrix into the formula. Here, \( a = 1-\lambda \), \( b = 3 \), \( c = 2 \), and \( d = 2-\lambda \). So the determinant is: \[ (1-\lambda)(2-\lambda) - (3 \times 2) \]

Step 4: Expand the Expression

Expand the expression \((1-\lambda)(2-\lambda)\): \[ (1-\lambda)(2-\lambda) = 1 \cdot 2 + (-\lambda) \cdot (-\lambda) + (-\lambda) \cdot 2 + 1 \cdot (-\lambda) \] This simplifies to: \[ = 2 + \lambda^2 - 2\lambda - \lambda \] \[ = \lambda^2 - 3\lambda + 2 \]

Step 5: Simplify the Entire Determinant

Now substitute the expanded expression back into the determinant formula: \[ \lambda^2 - 3\lambda + 2 - 6 \] Simplify this to: \[ \lambda^2 - 3\lambda - 4 \]

Step 6: State the Result

Thus, the determinant of the given matrix is \[ \lambda^2 - 3\lambda - 4 \].

Key Concepts

Eigenvalues

Eigenvalues are special numbers associated with a matrix. They offer insights into the matrix's characteristic polynomial. For square matrices like our 2x2 example, the eigenvalues are solutions to the equation where the determinant of the matrix minus lambda times the identity matrix equals zero.
Let's break this down further:

• Start by subtracting lambda from the main diagonal elements of our matrix, resulting in a new matrix.
• Next, find the determinant of this altered matrix.
• By setting this determinant to zero, we can solve for lambda, giving us the eigenvalues.

This process is fundamental in understanding matrix transformations as eigenvalues represent the scaling factors along the direction of the eigenvectors. If you think of the matrix as a transformation matrix, then the eigenvalues tell you how much that transformation stretches or compresses space in particular directions.

Linear Algebra

Linear algebra is a mathematical discipline focused on vector spaces and linear mappings between them. It's a foundational tool for understanding systems of linear equations, transformations, and their properties. In our context:

• Matrices are key tools in linear algebra as they represent linear transformations of space.
• A determinant provides a scalar value that conveys information about the volume transformation and orientation induced by a matrix.
• In simpler terms, if a matrix represents a function mapping one vector to another, its determinant indicates how space is expanded or contracted by that function.
• Zero determinants signal that the matrix is singular, meaning it squashes space into a lower dimension.

Linear algebra is crucial for various applications, such as in physics for modeling forces and in computer science for machine learning algorithms. Understanding matrices and determinants is essential for grasping how these transformations work.

Polynomial Expression

A polynomial expression is a mathematical sentence composed of variables and coefficients, arranged in terms of powers. Polynomials are pivotal in determining matrix characteristics. In the determinant expression of our 2x2 matrix:

• The expanded form \( (1-\lambda)(2-\lambda) - 6 \) results in a polynomial \( \lambda^2 - 3\lambda - 4 \).
• This polynomial is derived from the determinant calculation and serves as the characteristic polynomial of the matrix.
• The roots of this polynomial correspond to the eigenvalues, showcasing the interplay between polynomial theory and linear algebra.

Polynomials aid in expressing complex algebraic ideas succinctly. They also help in finding critical points in systems represented by matrices—like the scaling factors in transformations, as shown by our determinant result for the eigenvalues. Becoming comfortable with polynomial expressions is vital for solving higher-level math problems.