Problem 25 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{rr} 8 & 8 \\ -9 & -9 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Understand the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula \( ad - bc \). We will apply this formula to solve the given determinant.

Step 2: Identify Elements of the Matrix

Identify the values of \(a\), \(b\), \(c\), and \(d\) in the given matrix: \(a = 8\), \(b = 8\), \(c = -9\), and \(d = -9\). We will use these identified values in the next step.

Step 3: Plug Values into the Determinant Formula

Substitute the identified values into the formula: \( ad - bc \). Thus, we have \( 8(-9) - 8(-9) \).

Step 4: Calculate Each Multiplication

Calculate \( ad = 8 \times (-9) = -72 \) and \( bc = 8 \times (-9) = -72 \).

Step 5: Subtract the Results

Now, substitute the results from Step 4 into the formula: \(-72 - (-72) = -72 + 72 = 0\).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple and essential component in the study of matrices. This type of matrix has two rows and two columns, providing four elements. In this context, the matrix can be represented as:

• First row:
  • Element a (first column)
  • Element b (second column)
• Second row:
  • Element c (first column)
  • Element d (second column)

Each element in the matrix plays a crucial role in computations, such as determinant calculation. Remember, every element has a specific position, crucial for matrix operations.
Understanding this structure serves as the foundation for exploring more complex matrices in intermediate algebra.

Determinant Calculation Essentials

The determinant is a special number derived from a square matrix, like the 2x2 matrix. It provides valuable information about the properties of the matrix. To calculate the determinant of a 2x2 matrix, use the simple formula:\[|A| = ad - bc\]Where \(a\), \(b\), \(c\), and \(d\) are elements of the matrix positioned as:

• \(a\) and \(b\) in the first row
• \(c\) and \(d\) in the second row

Once these elements are identified, simply multiply \(a\) by \(d\) and \(b\) by \(c\), then subtract the second product from the first. This result gives the determinant, which can help understand various matrix behaviors, such as singularity or invertibility.

An Introduction to Matrix Algebra

Matrix algebra is the branch of mathematics that deals with matrices and their operations. It includes addition, subtraction, multiplication, and finding determinants. A basic understanding of matrices allows you to solve systems of linear equations effectively.
Matrix algebra is not just a theoretical concept—it's widely used in fields such as computer science, engineering, and physics. Understanding how to manipulate matrices opens up a host of possibilities in modeling real-world problems. The beauty of matrix calculations lies in their ability to transform sets of equations into manageable insights using basic algebraic operations.

Intermediate Algebra and Matrix Relevance

Intermediate algebra serves as a bridge between basic arithmetic and advanced math, providing the skills needed to solve practical problems. Within this context, matrices are introduced as powerful tools to handle linear equations and transformations. A 2x2 matrix, for example, can represent transformations in two-dimensional space, such as rotations or scalings.
In intermediate algebra, understanding the concept of a determinant is crucial, as it helps determine whether a matrix is invertible or singular. These concepts are necessary for solving higher-level problems in calculus and beyond, highlighting the importance of mastering matrix operations early on in your mathematical studies.