Problem 18 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{cc} -\frac{2}{3} & 10 \\ -\frac{1}{2} & 6 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 1.

Step 1: Set Up the Determinant Formula

The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) is given by the formula: \[det(A) = ad - bc \].For the given matrix, \(a = -\frac{2}{3}\), \(b = 10\), \(c = -\frac{1}{2}\), and \(d = 6\).

Step 2: Multiply the Diagonal Elements

Calculate the product of the diagonal elements: \[ad - bc = \left(-\frac{2}{3}\right)(6)\].This multiplication results in \(-\frac{2}{3} \times 6 = -4.\)

Step 3: Multiply the Off-Diagonal Elements

Calculate the product of the off-diagonal elements: \[bc = (10)\left(-\frac{1}{2}\right)\].This multiplication results in \(10 \times -\frac{1}{2} = -5.\)

Step 4: Subtract the Off-Diagonal Product from the Diagonal Product

Apply the determinant formula: \[ad - bc = -4 - (-5)\].Evaluate the subtraction: \(-4 + 5 = 1.\)

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a simple square matrix that consists of two rows and two columns. Each element within the matrix is represented by a letter or a number and positioned at specific coordinates, allowing us to perform various matrix operations. In this case, the 2x2 matrix looked like this:
\[\begin{bmatrix} -\frac{2}{3} & 10 \ -\frac{1}{2} & 6 \\end{bmatrix}\]This configuration lets us apply different mathematical procedures, like determining the determinant. The formula for the determinant of this type of matrix is straightforward, as it involves a simple calculation using its four elements. Understanding this matrix type is essential because it forms the foundation on which more complex matrices and operations are built. Simplifying these basic operations helps in comprehending more sophisticated matrix structures and their applications.

Matrix Operations Explained

Matrix operations form a critical part of linear algebra, allowing us to solve complex problems in fields ranging from computer science to physics. Two basic operations frequently performed on matrices include addition/subtraction and multiplication.

• Addition/Subtraction: These operations are performed by adding or subtracting corresponding elements of the matrices.
• Multiplication: Here, matrices are multiplied in a manner consistent with multiplication of numbers, but it involves each element of a row in the first matrix being multiplied by the corresponding element of a column in the second matrix. Specifically for 2x2 matrices, the determinant is calculated as a special form of multiplication, giving us insight into properties such as area scaling and matrix invertibility.

The determinant operation specifically highlights how certain matrix transformations affect geometric properties, making it an essential tool for analyzing linear transformations.

Introduction to Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. The study of determinants, like in the exercise, is a fundamental component of this field.

Understanding linear algebra is vital because it equips us with the tools to model real-world problems such as optimization in engineering, constructing networks, and understanding multi-dimensional spaces. Determinants, in particular, help in establishing the invertibility of matrices, solving systems of linear equations, and finding eigenvalues and eigenvectors. These concepts are powerful in predicting and analyzing trends in data or in physical systems.
Embedded in every linear algebra task is the principle of solving through transformations and operations, stepping from basic 2x2 matrices to vast and complex matrix configurations.