Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}6 & -4 \\\0 & -1\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is -6.

Step 1: Understanding the Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by the formula \( ad - bc \). For this problem, we need to identify the elements \(a, b, c, \) and \(d\).

Step 2: Identify Matrix Elements

In the matrix \( \begin{bmatrix} 6 & -4 \ 0 & -1 \end{bmatrix} \), the elements are as follows: \(a = 6\), \(b = -4\), \(c = 0\), and \(d = -1\).

Step 3: Apply the 2x2 Determinant Formula

Using the formula \( ad - bc \), substitute the identified elements: \(6\times(-1) - (-4)\times 0\).

Step 4: Simplify the Expression

Calculate \(6\times(-1) = -6\) and \(-4\times 0 = 0\), therefore the determinant is \( -6 - 0 = -6 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple mathematical structure consisting of two rows and two columns. It is often written in a form that looks like this: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\].
Such a matrix is a collection of four numbers or elements, represented as \(a, b, c,\) and \(d\). These numbers are arranged in two horizontal lines called rows and two vertical lines called columns.

2x2 matrices are fundamental in various fields such as physics, computer science, and engineering. They are useful for representing transformations, systems of equations, and data.

In practical applications, understanding and using 2x2 matrices can help solve problems involving rotations, reflections, and scaling, especially in computer graphics and geometry.

It's essential to get comfortable with reading and writing these simple matrices, as they are often the building blocks for learning more complex matrix operations.

Matrix Elements

Matrix elements are the individual numbers within a matrix. In the 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\], these elements are:

• \(a\) – the top-left element
• \(b\) – the top-right element
• \(c\) – the bottom-left element
• \(d\) – the bottom-right element

Recognizing these elements is crucial because each plays a specific role in calculations involving matrices, like determining a matrix's determinant.

In our example matrix \[\begin{bmatrix} 6 & -4 \ 0 & -1 \end{bmatrix}\], the elements are explicitly given as:

• \(a = 6\)
• \(b = -4\)
• \(c = 0\)
• \(d = -1\)

Understanding how to identify and manipulate these elements will simplify tasks such as performing operations and solving matrix-related problems.

Determinant Formula

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix, it provides a scalar value that can signal whether a set of linear equations has a unique solution, among other properties.

The formula for finding the determinant of a 2x2 matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] is: \[ ad - bc \]
This means you multiply the top-left element \(a\) by the bottom-right element \(d\), and then subtract the result of multiplying the top-right element \(b\) by the bottom-left element \(c\).

Let's put this into practice using our example. For the matrix \[\begin{bmatrix} 6 & -4 \ 0 & -1 \end{bmatrix}\], apply the formula:

• Multiply: \(6 \times -1 = -6\)
• Multiply: \(-4 \times 0 = 0\)
• Subtract: \(-6 - 0 = -6\)

So, the determinant of the matrix is \(-6\). This straightforward method of calculating determinants is foundational for more complex operations involving matrices in algebra and calculus.