/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 18 Evaluate each determinant. $$ ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 18

Evaluate each determinant. $$ \left|\begin{array}{rr} -1 & 2 \\ 3 & -4 \end{array}\right| $$

Short Answer

Expert verified
The determinant is -2.

Step by step solution

01

Identify the Determinant Structure

This is a 2x2 determinant, which has the structure \[ \left| \begin{array}{cc} a & b \ c & d \end{array} \right|. \] For this problem, \( a = -1 \), \( b = 2 \), \( c = 3 \), and \( d = -4 \).
02

Apply the Formula for 2x2 Determinants

The formula for calculating the determinant of a 2x2 matrix is \( ad - bc \). Substitute \( a = -1 \), \( b = 2 \), \( c = 3 \), and \( d = -4 \) into the formula.
03

Perform the Calculations

Calculate \( ad \) and \( bc \):\[ ad = (-1)(-4) = 4 \] and \[ bc = (2)(3) = 6. \] Now, substitute these into the determinant formula: \[ ad - bc = 4 - 6. \]
04

Simplify the Result

Simplify the expression: \[ 4 - 6 = -2. \] Thus, the determinant is \( -2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

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 grid of numbers with two rows and two columns. Here, each element in the matrix can be a number, a variable, or a more complex expression. For our exercise, the matrix looks like this:\[ \left| \begin{array}{rr} -1 & 2 \ 3 & -4 \end{array} \right| \] Understanding the structure of a 2x2 matrix is crucial for calculating its determinant. This setup allows us to apply specific methods, like the determinant formula specific to 2x2 matrices. Recognizing and properly arranging the elements \(a\), \(b\), \(c\), and \(d\) is the key to using the formula correctly.
The Determinant Calculation
Calculating the determinant of a 2x2 matrix involves using a straightforward formula. This formula helps us understand specific properties of the matrix, such as whether it is invertible or how linear transformations associated with the matrix behave. For a 2x2 matrix formatted as:\[ \left| \begin{array}{cc} a & b \ c & d \end{array} \right| \] The determinant \(D\) is calculated using:\[ D = ad - bc \]This formula takes the product of the "diagonals" of the matrix:
  • Multiply \(a\) and \(d\) for the product \(ad\)
  • Multiply \(b\) and \(c\) for the product \(bc\)
Then, the determinant is found by subtracting \(bc\) from \(ad\), which reflects the net effect of these interactions in the matrix.
Simplifying Algebraic Expressions
Once you've applied the determinant formula, the next step is to simplify the resulting algebraic expression. For our matrix, after substituting the values into the determinant formula, we find:
  • \(ad = (-1) \times (-4) = 4\)
  • \(bc = (2) \times (3) = 6\)
The expression for the determinant then becomes:\[ 4 - 6 \]This expression simplifies to \(-2\) by straightforward arithmetic subtraction. Simplifying algebraic expressions is key to ensuring the results are accurate and clear, providing you with the final value of the determinant, which is \(-2\) in this case.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. $$ \left\\{\begin{array}{l} 5 x+3 y=4 \\ 3 y-4 z=4 \\ x+z=1 \end{array}\right. $$

For the function \(y=2 x^{2}+6 x+1,\) what is the independent variable and what is the dependent variable?

What is the name of the function \(f(x)=|x| ?\)

Are the lines described by \(y=2 x-7\) and \(2 x-y=10\) parallel?

Explain the difference between a matrix and a determinant. Give an example of each.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.