Problem 13 - Advanced Engineering Mathematics

Question

In Problems 9-14, 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

The determinant of the matrix is \( \lambda^2 - 3\lambda - 4 \).

Step 1: Recall Determinant Formula for a 2x2 Matrix

For a matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by the formula \( ad - bc \). This is the basic formula that will be used to find the determinant of any 2x2 matrix.

Step 2: Identify Elements of the Matrix

Identify the elements of the given matrix \( \begin{pmatrix} 1-\lambda & 3 \ 2 & 2-\lambda \end{pmatrix} \). Here, \( a = 1-\lambda \), \( b = 3 \), \( c = 2 \), and \( d = 2-\lambda \).

Step 3: Apply the Determinant Formula

Substitute the identified elements into the determinant formula: \( (1-\lambda)(2-\lambda) - (3)(2) \). This translates to calculating \( (1-\lambda)(2-\lambda) - 6 \).

Step 4: Expand the Expression

Expand the expression \( (1-\lambda)(2-\lambda) \) using the distributive property: \( 1 \cdot 2 - 1 \cdot \lambda - \lambda \cdot 2 + \lambda^2 \), which simplifies to \( 2 - \lambda - 2\lambda + \lambda^2 \).

Step 5: Simplify the Expanded Terms

Combine like terms from the expansion: \( 2 - 3\lambda + \lambda^2 \).

Step 6: Finalize the Determinant Calculation

Subtract 6 from the simplified expression (from Step 5): \( 2 - 3\lambda + \lambda^2 - 6 = \lambda^2 - 3\lambda - 4 \).

Step 7: Write the Final Answer

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

Key Concepts

2x2 matrix

A 2x2 matrix is a simple yet fundamental type of matrix in mathematics, especially in linear algebra. It has 2 rows and 2 columns, forming a square-like configuration. This structure is often used because of its simplicity and the ease with which certain calculations, like determinants, can be performed.

In a 2x2 matrix, each element can be represented with a letter, forming the general matrix template:

• The top-left element is referred to as a.
• The top-right element is referred to as b.
• The bottom-left element is referred to as c.
• The bottom-right element is referred to as d.

Matrices are often used to represent systems of linear equations and transformations in space.
Understanding the 2x2 matrix layout is crucial because each part plays a role when calculating further properties, like the determinant.

matrix algebra

Matrix algebra involves operations similar to those in regular algebra, like addition and multiplication. However, since matrices consist of numbers arranged in rows and columns, distinct rules apply to them.
In the context of a 2x2 matrix, calculating the determinant is a significant aspect of matrix algebra. The determinant is a special number that you can calculate from a square matrix. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), its determinant is determined by the formula \( ad - bc \).
This operation has a specific meaning:

• It helps in determining whether a matrix is invertible (can be reversed).
• An invertible matrix has a non-zero determinant.
• If the determinant is zero, the matrix cannot be inverted.

Matrix algebra thus provides powerful tools for solving linear equations, transforming coordinates, and more.

distributive property

The distributive property is a fundamental concept not only in algebra but also when working with matrices. It dictates how multiplication over addition should be handled.
The distributive property states that for any numbers or expressions \( a, b, \) and \( c \), \ :

• \( a(b + c) = ab + ac \)

This rule significantly simplifies calculations, especially when expanding expressions involving variables.
In the exercise, applying the distributive property allowed us to expand the product \((1-\lambda)(2-\lambda)\) as follows:

• \( 1 \times 2 \) yields \( 2 \)
• \( 1 \times (-\lambda) \) yields \(-\lambda \)
• \(-\lambda \times 2 \) yields \(-2\lambda \)
• \(-\lambda \times (-\lambda) \) yields \(\lambda^2\)

This yields the expanded expression: \( 2 - 3\lambda + \lambda^2 \), simplifying any further calculations.

polynomial expansion

Polynomial expansion occurs when you multiply expressions to open up parentheses, showcasing each term separately.
In the particular exercise, we had the polynomial expression \((1-\lambda)(2-\lambda)\). Using the distributive property, we expanded this expression to \( 2 - \lambda - 2\lambda + \lambda^2 \). To expand a polynomial:

• Identify multiplication terms.
• Apply the distributive property for multiplication over addition.
• Combine like terms.

The last step involves combining terms that have the same variable powers, leading to the simplest form possible. In our example, \( -\lambda\) and \(-2\lambda\) were combined to \(-3\lambda\).
The expanded polynomial \( \lambda^2 - 3\lambda - 4 \) represents the determinant after considering every term systematically and illustrates the importance of careful expansion.