Problem 12 - College Algebra with Corequisite Support

Question

For the following exercises, find the determinant. $$ \left|\begin{array}{rr} 6 & -3 \\ 8 & 4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the matrix is 48.

Step 1: Identify Formula for Determinant of 2x2 Matrix

For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant \( \text{det}(A) \) is calculated as \( ad - bc \).

Step 2: Assign Values from Matrix

From the given matrix \( \begin{pmatrix} 6 & -3 \ 8 & 4 \end{pmatrix} \), assign values: \( a = 6 \), \( b = -3 \), \( c = 8 \), and \( d = 4 \).

Step 3: Substitute Values into Determinant Formula

Substitute the values into the determinant formula: \( 6 \times 4 - (-3) \times 8 \).

Step 4: Perform Multiplications

Calculate the multiplications: \( 6 \times 4 = 24 \) and \( (-3) \times 8 = -24 \).

Step 5: Calculate Final Determinant

Find the determinant by calculating \( 24 - (-24) = 24 + 24 = 48 \).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is a simple square arrangement of numbers consisting of two rows and two columns. You can often see it written like this: \[\begin{pmatrix} a & b \ c & d \end{pmatrix}\]Where each of the letters represents a number in a specific position. In this setup, \(a\) and \(b\) are numbers in the first row, while \(c\) and \(d\) belong to the second row.
Matrices are used in various fields including mathematics, physics, and computer science. A 2x2 matrix can represent simple systems of equations, transformations, or even networks. Understanding this type of matrix is fundamental to grasping more complex matrix operations.
These matrices are also the building blocks for bigger, more complex matrices. Getting a good grip on 2x2 matrices helps you with more intricate mathematical structures later on.

Basic Matrix Operations

Matrix operations are essential tools in mathematics and numerous scientific disciplines. These types of operations include addition, subtraction, and multiplication. However, one important operation for understanding the properties of a matrix is determining its determinant.
The operation used in our example, finding the determinant of a 2x2 matrix:

• We start with a formula: \( ad - bc \), where each letter represents the elements of the matrix.
• This operation is vital because the determinant gives information about the matrix, such as whether it is invertible (i.e., if its determinant is not zero, it can be inverted).

Matrix operations multiply and solve complex problems in areas like physics simulations, computer graphics, and many others. Understanding these operations makes complex calculations easier and more efficient.

Performing Mathematical Calculations

Carrying out mathematical calculations is the core of determining a matrix's determinant. Here's how it usually unfolds:

• Identify all components from the matrix. For example, from the matrix \( \begin{pmatrix} 6 & -3 \ 8 & 4 \end{pmatrix}\), you'll map \(a = 6\), \(b = -3\), \(c = 8\), and \(d = 4\).
• Substitute these values into the determinant formula \( ad - bc \).
• Conduct the multiplications: compute \(6 \times 4\) to get 24, and \(-3 \times 8\) to get -24.
• Finally, handle the arithmetic: determine the determinant by evaluating \(24 - (-24)\), which simplifies to \(24 + 24 = 48\).

Each step requires attention to detail. These calculations not only tell you about the matrix itself but also about its broader applicability in solving equations or performing complex transformations. Understanding this is key to mastering matrix-related problems.