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Chapter 5: Problem 13

Evaluate the given determinants. $$\left|\begin{array}{cc} 0.75 & -1.32 \\ 0.15 & 1.18 \end{array}\right|$$

Short Answer

Expert verified
The determinant of the matrix is 1.083.

Step by step solution

01

Identify the Formula

The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated using the formula: \( ad - bc \). In this problem, the matrix elements are \( a = 0.75 \), \( b = -1.32 \), \( c = 0.15 \), and \( d = 1.18 \).
02

Calculate the Product of Diagonal Elements

The first part of the formula is \( ad \). Calculate this by multiplying \( a = 0.75 \) by \( d = 1.18 \): \( ad = 0.75 \times 1.18 = 0.885 \).
03

Calculate the Product of Off-Diagonal Elements

The second part of the formula is \( bc \). Calculate this by multiplying \( b = -1.32 \) by \( c = 0.15 \): \( bc = -1.32 \times 0.15 = -0.198 \).
04

Substitute into the Determinant Formula

Now, substitute \( ad = 0.885 \) and \( bc = -0.198 \) into the determinant formula: \( ad - bc = 0.885 - (-0.198) \).
05

Simplify the Expression

Simplify the expression \( 0.885 - (-0.198) \) to \( 0.885 + 0.198 = 1.083 \). Thus, the determinant is \( 1.083 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding a 2x2 Matrix
A 2x2 matrix is a simple yet powerful mathematical concept. It consists of two rows and two columns, making it a compact and manageable way to represent data. In mathematics, a matrix is a rectangular array of numbers. For a 2x2 matrix, you can denote it as follows:
  • First row contains elements a and b
  • Second row contains elements c and d
This array is enclosed in a set of square brackets or vertical bars. Such a format allows easy application of mathematical operations like addition, subtraction, and multiplication.

Mathematically, a 2x2 matrix can be written as \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] The entries can be any numbers, including decimals and negatives as we see in the given exercise.
Diving into Matrices
Matrices are everywhere in mathematics, acting as fundamental building blocks in linear algebra. They allow for the representation of systems of linear equations and transformations in multidimensional space.
  • They can be of any size - ranging from tiny 1x1 matrices to infinitely large ones
  • Operations on matrices let us solve equations and perform complex calculations efficiently

Matrices are not only numbers in an organized box, they encapsulate data that model real-world phenomena. For example, in computer graphics, they manipulate the location of pixels while in statistics, matrices summarize data interactions.

In particular, the determinant of a matrix, specifically in 2x2 matrices, provides critical insights such as solving equations, understanding matrix invertibility, and calculating area and volume in geometry.
Importance in Mathematics Education
Understanding matrices, starting with simple 2x2 examples, is crucial in mathematics education. They provide foundational skills for more advanced topics in algebra and calculus.
  • Students learn to recognize patterns and solve systems of linear equations
  • Allows for exploration of transformational geometry

Matrices also serve as excellent problem-solving tools to develop logical thinking and analytical skills. The determinant, integral to their study, helps connect geometric properties to algebraic expressions. This bridges gaps between different areas of mathematics enabling students to appreciate the subject鈥檚 interconnected nature.

Developing a deep understanding of 2x2 matrices emphasizes critical thinking, supporting students in tackling more complex mathematical challenges as they advance in their education. It鈥檚 like building blocks of logic, laying strong foundations for their mathematical journey.

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Most popular questions from this chapter

Set up appropriate systems of equations. All numbers are accurate to at least two significant digits. A company budgets 750,000 dollars in salaries, hardware, and computer time for the design of a new product. The salaries are as much as the others combined, and the hardware budget is twice the computer budget. How much is budgeted for each?

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Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. In an election, candidate \(A\) defeated candidate \(B\) by 2000 votes. If \(1.0 \%\) of those who voted for \(A\) had voted for \(B, B\) would have won by 1000 votes. How many votes did each receive?

Set up appropriate systems of two linear equations and solve the systems algebraically. All data are accurate to at least two significant digits. There are two types of offices in an office building, and a total of 54 offices. One type rents for \(\$ 900 /\) month and the other type rents for \(\$ 1250 /\) month. If all offices are occupied and the total rental income is \(\$ 55,600 /\) month, how many of each type are there?
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