Problem 7 - Intermediate Algebra

Question

Evaluate each of the following determinants. $$ \left|\begin{array}{rr} 8 & -3 \\ 6 & 4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 50.

Step 1: Identify the formula for a 2x2 determinant

For a 2x2 matrix \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \), the determinant is calculated as \( ad - bc \). This formula will help us evaluate the given determinant.

Step 2: Assign values to variables

In the given matrix \( \left|\begin{array}{rr} 8 & -3 \ 6 & 4 \end{array}\right| \), assign \( a = 8, \ b = -3, \ c = 6, \ d = 4 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple and foundational element in matrix algebra. It consists of two rows and two columns. Each position in the matrix holds a specific value. For example, the matrix \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \) represents four values in a square arrangement. This compact form is incredibly useful in various areas of mathematics and engineering. Matrices can encode information, transformations, and relationships in a structured manner. Two main properties characterize a 2x2 matrix:

• Its order is 2. This means it has two rows and two columns, making it a square matrix.
• The elements \( a, b, c, \text{ and } d \) are the components that define the matrix's particular entries and their positions.

Understanding the structure and properties of a 2x2 matrix is essential when learning about larger matrices. It's the simplest form of a square matrix, providing a basis for understanding more complex operations.

Matrix Algebra

Matrix algebra is a powerful tool in mathematics for handling various operations involving matrices. It extends basic algebraic operations like addition, subtraction, multiplication, and finding inverses to matrices. These operations allow us to perform complex computations and transformations conveniently.

• Addition: This involves adding corresponding elements of two matrices of the same size. Each element in one matrix is added to the corresponding element in the other matrix.
• Multiplication: This is a bit different from regular number multiplication. The result is obtained by taking the dot product of rows and columns from the matrices being multiplied.
• Identity and Inverse: An identity matrix acts like the number 1 in matrix operations, while an inverse matrix is similar to division, reversing the effects of the matrix.

Understanding matrix algebra is crucial for fields like physics, computer science, and statistics. It equips learners with the skills needed to manipulate matrices efficiently.

Determinant Formula

The determinant of a matrix is a special number that can be calculated from its elements. It's particularly important in solving systems of linear equations, understanding matrix invertibility, and analyzing matrix properties.For a 2x2 matrix, the determinant provides a straightforward calculation. Let's consider a matrix \( \left|\begin{array}{cc} a & b \ c & d \end{array}\right| \). The determinant formula for this matrix is given by:\[det(A) = ad - bc\]This formula involves:

• Multiplying the top-left element \( a \) with the bottom-right element \( d \).
• Then, subtracting the product of the top-right element \( b \) and the bottom-left element \( c \).

For example, in the original step-by-step solution, if you plug the values \( a = 8, b = -3, c = 6, \) and \( d = 4 \) into the formula, you'll get the determinant value of \( 8 \times 4 - (-3) \times 6 \), which results in 50. This calculation helps understand not just the number itself but the properties and potential uses of the matrix in further computations.