Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} -1.1 & 0.6 \\ 7.2 & -0.5 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is -3.77.

Step 1: Identifying Matrix Type

This is a 2x2 matrix, specifically:\[ A = \begin{bmatrix} -1.1 & 0.6 \ 7.2 & -0.5 \end{bmatrix} \] The determinant for a 2x2 matrix is calculated using a straightforward formula.

Step 2: Applying the Determinant Formula

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as:\[ \text{det}(A) = ad - bc \]In this matrix, \( a = -1.1 \), \( b = 0.6 \), \( c = 7.2 \), and \( d = -0.5 \).

Step 3: Calculating Terms in Formula

Calculate the products for the determinant formula:1. Calculate \( ad = (-1.1) \times (-0.5) = 0.55 \).2. Calculate \( bc = (0.6) \times (7.2) = 4.32 \).

Step 4: Finding the Determinant

Subtract the second product from the first to find the determinant:\[ \text{det}(A) = 0.55 - 4.32 = -3.77 \]

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with matrices and the operations that can be performed on them. This area of study is essential for various fields like physics, engineering, computer science, and more, as it provides a way to represent and solve linear equations.

In matrix algebra, a matrix is essentially a collection of numbers, arranged into a fixed number of rows and columns. Matrices can be of any dimension, such as 2x2, 3x3, etc. The 2x2 matrix from our exercise is among the most basic forms, easy to handle but powerful enough for many applications.

Matrices can undergo several operations including addition, subtraction, and multiplication, just like numbers. However, operations follow specific rules distinct from those used in standard arithmetic. Understanding these rules through matrix algebra is crucial as it serves as a foundation for more complex topics in linear algebra.

Determinant Calculation

The determinant is a special number associated with a square matrix. In a 2x2 matrix, the determinant gives you a single value from the elements of the matrix, which can tell you things about the matrix such as whether the matrix has an inverse or how it transforms space.

To calculate the determinant of a 2x2 matrix \[ \begin{bmatrix} a & b \c & d \ \end{bmatrix} \], use the formula: \( ext{det}(A) = ad - bc \)

This means we multiply the diagonal elements, subtract the product of the off-diagonal elements, and the result is the determinant. This simple calculation is foundational in linear algebra, providing a stepping stone to understand more complex matrix operations.

Linear Algebra Concepts

Linear algebra is a central branch of mathematics that covers vector spaces, linear mappings, and systems of linear equations. It extends beyond the operations on individual numbers, allowing us to study and solve problems using matrices and determinants.

One crucial aspect of linear algebra is its ability to represent real-world systems and transformations using matrices. For instance, matrices are used to describe transformations such as rotations, translations, and scalings in graphics and geometric modeling.

Determinants play a key role in understanding these transformations. For example, a nonzero determinant indicates that the matrix is invertible, which ensures that transformations can be reversed. This helps in analyzing and solving linear systems, making linear algebra a powerful tool in various scientific and engineering disciplines.