Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}-1 & 3 \\\\-2 & 9\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the given matrix is -3.

Step 1: Identify the Determinant Formula for a 2x2 Matrix

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula: \( \text{det} = ad - bc \). This means you need to take the product of the diagonals and then subtract their opposite.

Step 2: Assign Values to Variables

Identify the entries of the matrix. In this matrix, \( a = -1 \), \( b = 3 \), \( c = -2 \), and \( d = 9 \).

Step 3: Calculate the Product of the Main Diagonal Entries

Multiply \( a \) and \( d \): \( (-1) \times 9 = -9 \).

Step 4: Calculate the Product of the Off-Diagonal Entries

Multiply \( b \) and \( c \): \( 3 \times (-2) = -6 \).

Step 5: Calculate the Determinant Using the Formula

Apply the formula \( ext{det} = ad - bc \): substituting the values gives \( ext{det} = (-9) - (-6) \).

Step 6: Simplify the Expression

The expression becomes \( -9 + 6 = -3 \).

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix in mathematics. It is a rectangular array of numbers arranged in two rows and two columns. Here is a basic example:

• The top row contains the elements \( a \) and \( b \).
• The bottom row contains the elements \( c \) and \( d \).

The general form looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]These letters represent real numbers, and you can think of this matrix as a compact way to store four numerical values.
This structure is helpful in various calculations, such as solving systems of equations, transformations in geometry, and representing linear functions in algebra. By having a predefined format, matrices like the 2x2 make mathematical operations clearer and more organized.

Basics of Matrix Multiplication

Matrix multiplication is a method used for combining matrices and is different from simple arithmetic multiplication. For a 2x2 matrix, multiplicatively combining with another 2x2 matrix is done in a systematic manner:

• Each element of the resultant matrix is calculated by taking the dot product of the rows from the first matrix and the columns from the second matrix.
• Start by multiplying corresponding entries and add them up.

So if you multiply two matrices, each with their own elements \( a, b, c, \) and \( d \), you can determine the resulting matrix by applying this technique to each position.
It is worth noting that unlike regular multiplication, matrix multiplication is not commutative. This means that the order in which you multiply matrices matters, i.e., \( AB eq BA \). Understanding this concept is key when dealing with matrices, especially in more complex operations.
Matrix multiplication is foundational in vector space transformations, computer graphics, and is essential in many applications across science and engineering.

Importance of Mathematics Education

Mathematics education aims to provide students with essential quantitative and analytical skills. Understanding concepts like matrices and determinants is crucial because:

• It enhances problem-solving abilities, enabling students to tackle abstract and real-life problems.
• Mathematical thinking aids in logical reasoning and the ability to think critically.
• The tools and techniques used in mathematics are vital for several fields, including engineering, physics, computer science, and economics.

Learning math through practical examples, like solving determinants of matrices, helps in grasping the practical side of mathematical theories. It also provides the capability to model and predict outcomes in various professional tasks. By fostering these skills through mathematics education, students are better equipped to engage in a digital and technology-driven world, preparing them for future educational pursuits or career paths.