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Chapter 9: Problem 4

Find the value of each determinant. $$\left|\begin{array}{cc} 6 & -4 \\ 0 & -1 \end{array}\right|$$

Short Answer

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Step by step solution

01

Identify the formula for a 2x2 determinant

The determinant of a 2x2 matrix \textbf{A} = \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is calculated using the formula det(\textbf{A}) = ad - bc.
02

Assign values from the given determinant

For the matrix \( \begin{pmatrix} 6 & -4 \ 0 & -1 \end{pmatrix} \), identify a, b, c, and d: a = 6 b = -4 c = 0 d = -1.
03

Apply the determinant formula

Substitute the identified values into the determinant formula: det = (6)(-1) - (0)(-4).
04

Simplify the expression

Compute the value: det = -6 - 0 = -6.

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Key Concepts

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

2x2 Matrices
Matrices are a fundamental concept in linear algebra. A 2x2 matrix, specifically, is one in which there are 2 rows and 2 columns. These are often written in the following form:

\( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

Here, \(a, b, c, \text{and} d \) are the elements or entries of the matrix. Matrices are used for many purposes, such as solving systems of linear equations, transforming geometric data, and much more. Understanding how to handle simple 2x2 matrices will open the door to more complex topics in linear algebra.
Matrix Determinant Formula
The determinant of a matrix is a special number that helps us understand some properties of the matrix, such as whether it is invertible (non-singular). For a 2x2 matrix, it鈥檚 straightforward to calculate the determinant using a specific formula:
  • Given a 2x2 matrix \(A \): \( \begin{pmatrix} a & b \ c & d \end{pmatrix}\)
  • The determinant, denoted as \( \text{det}(A) \), is calculated as follows:

\[ \text{det}(A) = ad - bc \]

In simpler terms, multiply the top-left element (\(a\)) by the bottom-right element (\(d \)), then subtract the product of the top-right element (\(b \)) and the bottom-left element (\(c \)). This value helps us make important decisions about the matrix, such as determining solvability of linear systems.
Linear Algebra Basics
Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear mappings between those spaces. It has widespread applications, from science and engineering to computer science and economics. Here are some core concepts you need to get started:
  • Vectors: Objects that have both magnitude and direction.
  • Matrices: Rectangular arrays of numbers representing a set of linear equations.
  • Determinants: Scalars that provide essential information about a matrix and its properties.
  • Eigenvalues and Eigenvectors: Values and vectors that reveal fundamental characteristics of a matrix.

These basics create a foundation upon which you can build a deeper understanding of more complex topics, such as transformations and dimensional theory. Mastering these, especially determinants and matrices, is crucial for tackling higher-level problems in linear algebra.

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Most popular questions from this chapter

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Find the demand for the electric can opener at each price. (a) \(\$ 6\) (b) \(\$ 11\) (c) \(\$ 16\)

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rr} 0 & -5 \\ -4 & 2 \end{array}\right], B=\left[\begin{array}{rr} 3 & -1 \\ -5 & 4 \end{array}\right]$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} -\frac{3}{4} x+\frac{2}{3} y &=16 \\ \frac{5}{2} x+\frac{1}{2} y &=-37 \end{aligned}$$

Solve each problem. In certain parts of the Rocky Mountains, deer provide the main food source for mountain lions. When the deer population is large, the mountain lions thrive. However, a large mountain lion population reduces the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$ \left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{rr} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate at which the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 yr? 2 yr? (c) Consider part (b) but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of 1.01

Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((2,3),(-1,0),\) and \((-2,2)\) Find the equation of the parabola \(y=a x^{2}+b x+c\) that passes through the points \((-2,4),(2,2),\) and \((4,9)\)
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