Problem 3 - A Graphical Approach to College Algebra

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-1 & -2 \\\5 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 7.

Step 1: Understand the Determinant Formula for 2x2 Matrix

Recall that for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). In this case, \( a = -1 \), \( b = -2 \), \( c = 5 \), and \( d = 3 \).

Step 2: Substitute Values into the Formula

Substitute the given values into the determinant formula: \((-1) \times 3\) and \((-2) \times 5\).

Step 3: Perform the Multiplications

Calculate \((-1) \times 3 = -3\) and \((-2) \times 5 = -10\).

Step 4: Apply the Formula

Plug these results into the formula: the determinant is \(-3 - (-10)\).

Step 5: Compute the Final Result

Solve \(-3 + 10\), which gives a determinant of \(7\).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple square array comprising two rows and two columns. This kind of matrix is a foundational structure in linear algebra and is often the starting point for more complex matrix operations. Here's what a standard 2x2 matrix looks like:

• The top row contains elements labeled as "a" and "b"
• The bottom row includes elements "c" and "d"

This configuration can be written as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] Each element, denoted by these letters, plays a crucial role in various calculations, especially when it comes to finding determinants. Understanding how these elements work together is vital for mastering matrix algebra.

Matrix Algebra

Matrix algebra involves various operations using matrices, such as addition, subtraction, multiplication, and finding determinants. These operations allow us to transform and analyze data efficiently.
In the context of a 2x2 matrix, algebraic operations are usually straightforward to apply. For example:

• Addition and Subtraction: Simply add or subtract corresponding elements in the matrices. Ensure they are of the same size.
• Multiplication: This involves row-by-column multiplication, which is more complex but crucial for deep analysis.
• Determinant Calculation: The determinant provides meaningful insights about the matrix, indicating whether it is invertible and its geometrical properties.

These operations form the backbone of matrix algebra, facilitating numerous applications in science, engineering, and computer science.

Determinant Calculation

The determinant of a matrix is a scalar value that conveys important properties about the matrix. For any 2x2 matrix, the determinant calculation is straightforward and follows a specific formula.
Given a matrix: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]The determinant is defined by the formula:\[ad - bc\]This formula calculates quickly and provides essential insights into the matrix's behavior. Let's apply this to our example matrix:

• Elements: For \(a = -1\), \(b = -2\), \(c = 5\), and \(d = 3\)
• Multiplication: Compute \((-1) \times 3 = -3\) and \((-2) \times 5 = -10\)
• Substitute: Insert into the formula to get: \(-3 - (-10)\)
• Result: Solve to find the determinant is \(7\)

Calculating determinants helps determine the invertibility of a matrix, a useful concept in solving systems of equations and in various transformations.