Problem 2 - A Graphical Approach to College Algebra

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 is -3.

Step 1: Understand the formula for a 2x2 determinant

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

Step 2: Identify the values in the matrix

Identify \( a = -1 \), \( b = 3 \), \( c = -2 \), and \( d = 9 \) from the matrix \( \begin{bmatrix} -1 & 3 \ -2 & 9 \end{bmatrix} \).

Step 3: Substitute values into the determinant formula

Using the determinant formula \( ad - bc \), substitute: \((-1)(9) - (3)(-2)\).

Step 4: Calculate each part separately

First calculate \((-1)(9) = -9\), and then calculate \((3)(-2) = -6\).

Step 5: Solve the expression for the determinant

Combine the results: \(-9 - (-6) = -9 + 6 = -3\).

Key Concepts

2x2 Matrix

A 2x2 matrix, in its simplest form, is a compact and organized array with two rows and two columns. It looks something like this:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]In this matrix:

• The first row consists of elements \(a\) and \(b\).
• The second row consists of elements \(c\) and \(d\).

These elements can be numbers, expressions, or even symbols that represent specific quantities.
The beauty of the 2x2 matrix lies in its simplicity and its wide range of applications, from solving systems of linear equations to representing transformations in graphics.
Understanding the arrangement and significance of each element in the matrix is essential as it forms the basis for further mathematical operations and analyses.

Determinant Formula

The determinant of a 2x2 matrix provides significant insight into the properties of the matrix. It's calculated using a simple yet effective formula:\[\text{det}(A) = ad - bc\]Here, \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix \(\begin{bmatrix}a & b \c & d\end{bmatrix}\).

• \(ad\) is the product of the diagonal elements \(a\) and \(d\).
• \(bc\) is the product of the non-diagonal elements \(b\) and \(c\).
• The determinant is found by subtracting \(bc\) from \(ad\).

This calculation might seem straightforward, but it carries a deeper meaning.
If the determinant is zero, it indicates the matrix does not have an inverse, and the transformation it represents is not reversible. On the other hand, a nonzero determinant suggests that the matrix is invertible, and the corresponding transformation preserves distinct positions.
Mastering the determinant formula is crucial for analyzing and understanding the characteristics of matrices.

Matrix Mathematics

Matrix mathematics serves as a cornerstone in several branches of mathematics and applied sciences. At its core, it deals with arrays like our 2x2 matrix and employs operations such as addition, subtraction, multiplication, and finding determinants.
Matrices can represent and solve problems involving multiple variables with ease.

• They are utilized in systems of equations, which may appear daunting without the organized structure that matrices provide.
• In computer graphics, matrices transform and manipulate images and shapes.
• In physics, matrices describe movements and changes in mechanics and quantum state behaviors.

The calculation of determinants, as seen in the determinant formula, is an essential part of matrix mathematics. It allows us to find solutions to matrix equations and determine properties like invertibility.
By understanding matrix mathematics, we develop the skills to tackle complex mathematical problems efficiently and elegantly.