Problem 14 - Advanced Engineering Mathematics

Question

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} -3-\lambda & -4 \\ -2 & 5-\lambda \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant is \(\lambda^2 - 2\lambda - 23\).

Step 1: Define the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \(\left(\begin{array}{cc} a & b \ c & d \end{array}\right)\) is calculated using the formula \(ad - bc\). This formula will be used to evaluate the determinant of the given matrix.

Step 2: Identify the Elements of the Matrix

For the matrix \(\left(\begin{array}{cc} -3-\lambda & -4 \ -2 & 5-\lambda \end{array}\right)\), \(a = -3-\lambda\), \(b = -4\), \(c = -2\), and \(d = 5-\lambda\). These values will be substituted into the determinant formula.

Step 3: Substitute Values into the Determinant Formula

Using the formula from Step 1, substitute the identified values: \((-3-\lambda)(5-\lambda) - (-4)(-2) = ad - bc\).- Start with \((a)(d)\): \((-3-\lambda)(5-\lambda)\).- Then \(bc\): \((-4)(-2)\).

Step 4: Perform the Multiplication

Simplify \((-3-\lambda)(5-\lambda)\) using the distributive property: - \((-3)(5) = -15\)- \((-3)(-\lambda) = 3\lambda\)- \((-\lambda)(5) = -5\lambda\)- \((-\lambda)(-\lambda) = \lambda^2\)This results in: \(\lambda^2 - 5\lambda + 3\lambda - 15 = \lambda^2 - 2\lambda - 15\).The calculation of \((-4)(-2)\) yields \(8\).

Step 5: Calculate the Determinant

Plug the results from Step 4 back into the formula: \(\lambda^2 - 2\lambda - 15 - 8\).Simplify the expression:\(\lambda^2 - 2\lambda - 23\).

Step 6: Present the Final Answer

The determinant of the matrix is: \(\lambda^2 - 2\lambda - 23\).

Key Concepts

Determinant of a Matrix

The determinant is a key concept in linear algebra with several important applications. It is a scalar value that can be derived from a square matrix and provides insights into the matrix's properties. The determinant:

• Measures the 'volume' that the matrix scales in linear transformations.
• Determines whether a matrix is invertible. If the determinant is zero, the matrix does not have an inverse.
• Assists in solving system of linear equations, where a non-zero determinant suggests the equations have a unique solution.

Determinants can be complex to calculate for larger matrices, but for a 2x2 matrix, the computation is straightforward: take the product of the diagonal elements and subtract the product of the off-diagonal elements.

2x2 Matrix

The 2x2 matrix is one of the simplest forms within matrix algebra. It consists of four elements, arranged in two rows and two columns, typically denoted as:\[\left( \begin{array}{cc} a & b \ c & d \end{array} \right)\]This small size makes understanding the core properties and functions accessible.

• Simplicity: Computation such as determinant and trace is easier, which serves as a strong foundation for learning more complex matrix algebra.
• Transformation in 2D: A 2x2 matrix can represent linear transformations like rotations, reflections, and scaling in a two-dimensional space.
• Practical Applications: Used in systems of linear equations (two variables) and analyzing planar graphs.

In our exercise, a 2x2 matrix was provided with variables as elements, prompting the need for symbolic computation in finding its determinant.

Matrix Algebra

Matrix algebra is a powerful tool in mathematics and engineering, encompassing operations such as addition, subtraction, multiplication, and inversion of matrices. Here are key attributes:

• Linear Equations: Matrix algebra simplifies solving multiple linear equations simultaneously.
• Transformations: Used in graphics and data science to perform transformations and projections.
• Eigenvalues and Eigenvectors: Fundamental in simplifying matrix operations and understanding systems behavior.

In the context of the exercise, understanding matrix algebra was essential for manipulating and simplifying the 2x2 matrix's elements to compute the determinant. This also illustrated how symbolic computations, with elements like \(-3-\lambda\), could be handled effectively through matrix operations.