Problem 3 - Algebra for College Students

Question

Evaluate each \(2 \times 2\) determinant by using Definition 11.1. \(\left|\begin{array}{rr}-3 & 2 \\ 7 & 5\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is -29.

Step 1: Apply the Formula for a 2x2 Determinant

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is found using the formula \( ad - bc \). For the given matrix \( \begin{bmatrix} -3 & 2 \ 7 & 5 \end{bmatrix} \), identify \( a = -3 \), \( b = 2 \), \( c = 7 \), and \( d = 5 \).

Step 2: Substitute the Values into the Formula

Substitute the identified values into the formula: \( (-3) \times 5 - 2 \times 7 \).

Step 3: Perform the Multiplications

Calculate the products: \( (-3) \times 5 = -15 \) and \( 2 \times 7 = 14 \).

Step 4: Calculate the Difference

Subtract the second product from the first: \( -15 - 14 = -29 \). This is the value of the determinant.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a simple form of a matrix, which is a rectangular array of numbers arranged in two rows and two columns. These matrices are essential elements in linear algebra, providing a foundation for many operations and applications. In a 2x2 matrix, elements are typically denoted as

• \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

where \(a, b, c,\) and \(d\) are numerical values.

The 2x2 matrix serves as a fundamental building block for larger matrices and plays a key role in solving systems of equations, transforming geometric shapes, and more.

Each number within the matrix is referred to as an element, and they have specific positions identified by their row and column numbers. Understanding these small size matrices is crucial because their simplicity helps in grasping more complex concepts of matrix operations.

Matrix Determinants Explained

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix, the determinant provides information about the matrix's properties, such as whether it is invertible or not. To find the determinant of a matrix, use the formula:

• \[det(\begin{bmatrix} a & b \ c & d \end{bmatrix}) = ad - bc\]

This formula only applies to 2x2 matrices, higher-order matrices require more complex calculations.

Determinants have several applications including solving linear equations, calculating areas and volumes, and describing the scaling factor of linear transformations.

For our exercise, substituting the given values \(a = -3, b = 2, c = 7,\) and \(d = 5\) into the formula results in the determinant value of -29.

This negative value indicates certain properties of the transformation that the matrix describes. Understanding determinants is crucial for deeper comprehension of matrix behaviors in various mathematical and physical contexts.

Basics of Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors and matrices and the operations that can be carried out with them.

It is fundamental in areas like engineering, physics, computer science, and economics, bridging the gap between mathematical theory and practical application. In linear algebra, matrices like the 2x2 matrix discussed earlier are used to represent linear transformations. Determinants are a key tool in determining the solvability of systems of linear equations.

• Vectors represent quantities having both magnitude and direction.
• Matrices represent arrays of vectors or coefficients in linear equations.

Understanding these concepts allows us to solve equations involving multiple unknowns, perform computer graphics transformations, and even encrypt data securely.
Learning linear algebra opens up a world of mathematical possibilities and practical applications that are foundational to advanced studies and professional work in many fields. This exercise of evaluating the determinant helps build a step towards mastering these expansive topics.