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Chapter 3: Problem 16

$$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right] $$

Short Answer

Expert verified
The determinant of the given upper triangular matrix is \(1\). To find the determinant, simply multiply the entries on the main diagonal: \(1 \times 1 \times 1 = 1\).

Step by step solution

01

Write down the given matrix

Note down the given matrix, which is an upper triangular matrix: $$ \left[\begin{array}{lll} 1 & 2 & 3 \\\ 0 & 1 & 2 \\\ 0 & 0 & 1 \end{array}\right] $$
02

Identify the format of the matrix

Observe that the given matrix has zeroes below the main diagonal, which means that it is an upper triangular matrix.
03

Calculate the determinant of an upper triangular matrix

To find the determinant of an upper triangular matrix, simply multiply the entries on the main diagonal: $$ \begin{aligned} \text{Determinant} &=1\times 1\times 1\\ &= 1 \end{aligned} $$
04

Write down the final answer

The determinant of the given upper triangular matrix is 1.

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Key Concepts

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

Understanding Upper Triangular Matrices
An upper triangular matrix is a special type of square matrix. In such a matrix, all the elements below the main diagonal are zero. This structure greatly simplifies various mathematical operations, like finding determinants or solving linear equations.

Key characteristics of upper triangular matrices include:
  • All elements below the main diagonal are zero.
  • The non-zero elements are found on and above the main diagonal.
  • It is always a square matrix, meaning it has the same number of rows and columns.
Working with upper triangular matrices can make calculations easier. Especially when it comes to the determinant, which can be found by simply multiplying the diagonal elements. This property can save time and computational effort compared to regular matrices.
Exploring the Main Diagonal
The main diagonal of a matrix is made up of the elements that stretch from the top left to the bottom right. These are the entries where the row number equals the column number.

In the case of an upper triangular matrix, the main diagonal plays a critical role, as it contains all the necessary non-zero elements needed to compute the determinant. For instance, in the matrix:
\[ \left[\begin{array}{lll} 1 & 2 & 3 \0 & 1 & 2 \0 & 0 & 1 \end{array}\right] \]
the main diagonal is composed of 1, 1, and 1. Understanding the main diagonal helps in simplifying many matrix-related computations.

To find the determinant of an upper triangular matrix, focus solely on these diagonal elements, underscoring their significance in matrix mathematics.
Matrix Mathematics Made Simple
Matrix mathematics involves operations and concepts applied to grid-like arrays, called matrices. This branch of mathematics is fundamental in various fields, including computer science, physics, and engineering.

When dealing with matrices, understanding matrix types like upper triangular matrices can simplify complex calculations.
  • Determinants: For an upper triangular matrix, the determinant can be simply calculated by multiplying the diagonal elements, streamlining the process.
  • Equations: Upper triangular matrices make solving systems of equations straightforward due to their structure.
Matrix mathematics allows us to solve linear equations, perform transformations, and much more. Grasping the basics, like identifying types of matrices and their properties, can make these tasks far more approachable.

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Most popular questions from this chapter

The Prisoner's Dilemma Slim Lefty and Joe Rap have been arrested for grand theft auto, having been caught red-handed driving away in a stolen 2010 Porsche. Although the police have more than enough evidence to convict them both, they feel that a confession would simplify the work of the prosecution. They decide to interrogate the prisoners separately. Slim and Joe are both told of the following plea-bargaining arrangement: If both confess, they will each receive a 2 -year sentence; if neither confesses, they will both receive five-year sentences, and if only one confesses (and thus squeals on the other), he will receive a suspended sentence, while the other will receive a 10 -year sentence. What should Slim do?

More Retail Discount Wars Your Abercrom B men's fashion outlet has a \(30 \%\) chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a \(70 \%\) chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a \(20 \%\) chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: Abercrom \(\mathbf{A}\) \(\begin{array}{ll}\text { Torn Shirts } & \text { Empty Shirts }\end{array}\) Mechanics \(\left[\begin{array}{rr}10 & -40 \\ -30 & 50\end{array}\right]\) Abercrom B \(\quad\) Military What is the expected resulting effect on your customer base?

Evaluate the given expression. Take$$\begin{aligned}&A=\left[\begin{array}{rr}0 & -1 \\\1 & 0 \\\\-1 & 2 \end{array}\right], B=\left[\begin{array}{rr}0.25 & -1 \\\0 & 0.5 \\\\-1 & 3\end{array}\right], \text { and } \\\&C=\left[\begin{array}{rr}1 & -1 \\\1 & 1 \\\\-1 & -1\end{array}\right].\end{aligned}$$ $$ A-C $$

Evaluate the given expression. Take \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\\ 0 & 2 & -1\end{array}\right], B=\left[\begin{array}{rrr}3 & 0 & -1 \\ 5 & -1 & 1\end{array}\right]\), and \(C=\left[\begin{array}{lll}x & 1 & w \\ z & r & 4\end{array}\right] .\) $$ 2 A-B $$

I Population Movement In 2006, the population of the United States, broken down by regions, was \(55.1\) million in the Northeast, \(66.2\) million in the Midwest, \(110.0\) million in the South, and \(70.0\) million in the West. \({ }^{19}\) The matrix \(P\) below shows the population movement during the period 2006 \(2007 .\) (Thus, \(98.92 \%\) of the population in the Northeast stayed there, while \(0.17 \%\) of the population in the Northeast moved to the Midwest, and so on.) \(\begin{array}{ccll}\text { To } & \text { To } & \text { To } & \text { To } \\\ \text { NE } & \text { MW } & \text { S } & \text { W }\end{array}\) \(P=\begin{aligned}&\text { From NE } \\\&\text { From MW } \\\&\text { From S } \\\&\text { From W }\end{aligned}\left[\begin{array}{llll}0.9892 & 0.0017 & 0.0073 & 0.0018 \\ 0.0010 & 0.9920 & 0.0048 & 0.0022 \\ 0.0018 & 0.0024 & 0.9934 & 0.0024 \\ 0.0008 & 0.0033 & 0.0045 & 0.9914\end{array}\right]\) Set up the 2006 population figures as a row vector. Assuming that these percentages also describe the population movements from 2005 to 2006 , show how matrix inversion and multiplication allow you to compute the population in each region in 2005 . (Round all answers to the nearest \(0.1\) million.)
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