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Chapter 3: Problem 4

Find the dimensions of the given matrix and identify the given entry. $$ D=\left[\begin{array}{rr} 15 & -18 \\ 6 & 0 \\ -6 & 5 \\ 48 & 18 \end{array}\right] ; d_{31} $$

Short Answer

Expert verified
The dimensions of the matrix D are 4x2. The entry \(d_{31}\) is located at the third row and the first column, so \(d_{31} = -6\).

Step by step solution

01

Find the dimensions of the matrix

To find the matrix dimensions, count the number of rows and columns. Rows are horizontal, and columns are vertical. In the given matrix D: $$ D= \left[\begin{array}{rr} 15 & -18 \\ 6 & 0 \\ -6 & 5 \\ 48 & 18 \end{array}\right] $$ There are 4 rows and 2 columns in this matrix. So, the dimensions of matrix D are 4x2.
02

Identify the entry d_{31} from the matrix

The given entry is d_{31}. The first numeral represents the row number and the second numeral represents the column number. Therefore, d_{31} is the element located at the third row and the first column. Now let's find the entry: $$ D= \left[\begin{array}{rr} 15 & -18 \\ 6 & 0 \\ -6 & 5 \\ 48 & 18 \end{array}\right] $$ Go to the third row and the first column, and you find the entry -6. So, d_{31} = -6.

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Key Concepts

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

Understanding Matrices
Matrices are fundamental components in mathematics. You can think of them as a grid, where numbers are arranged in rows and columns. These grids are important in various fields such as computer science and engineering. When dealing with matrices, it's crucial to be able to identify its dimensions and specific elements.
  • Matrices can store data in a structured format.
  • They are used to perform operations such as addition, multiplication, and inversion.
  • Matrices are defined by the number of rows and columns, forming their dimension.
Understanding how to navigate through this grid and manipulate the elements is a key skill in many mathematical applications.
Matrix Rows and Columns
When working with matrices, it is important to distinguish between rows and columns to understand matrix dimensions. In a matrix, rows run horizontally from left to right, while columns run vertically from top to bottom.
  • A row is a single sequence of elements that extends horizontally.
  • A column is a sequence of elements that rises vertically through the matrix.
  • The dimensions of a matrix are described by the number of rows and columns, written as rows x columns. For example, a matrix with 4 rows and 2 columns has dimensions of 4x2.
Recognizing rows and columns helps in performing tasks such as identifying specific elements in a matrix.
Identifying Matrix Elements
Locating a specific element within a matrix involves using indices that refer to its position in the grid. These indices follow a specific format, which involves row and column numbers.
  • The format is typically noted as \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number.
  • For example, \(a_{31}\) refers to the element in the third row and first column of a matrix.
  • In a 4x2 matrix, such as Matrix \(D\) in our example, navigate to the third row and check the first column to find the element \(-6\).
Using this method allows you to effectively read and interpret matrices, especially when solving mathematical problems or organizing data.

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Most popular questions from this chapter

Microbucks Computer Company makes two computers, the Pomegranate II and the Pomegranate Classic, at two different factories. The Pom II requires 2 processor chips, 16 memory chips, and 20 vacuum tubes, while the Pom Classic requires 1 processor chip, 4 memory chips, and 40 vacuum tubes. Microbucks has in stock at the beginning of the year 500 processor chips, 5,000 memory chips, and 10,000 vacuum tubes at the Pom II factory, and 200 processor chips, 2,000 memory chips, and 20,000 vacuum tubes at the Pom Classic factory. It manufactures 50 Pom II's and 50 Pom Classics each month. a. Find the company's inventory of parts after two months, using matrix operations. b. When (if ever) will the company run out of one of the parts?

Multiple Choice: If \(A\) and \(B\) are square matrices with \(A B=I\) and \(B A=I\), then (A) \(B\) is the inverse of \(A\). (B) \(A\) and \(B\) must be equal. (C) \(A\) and \(B\) must both be singular. (D) At least one of \(A\) and \(B\) is singular.

What would it mean if the total output figure for a particular sector of an input-output table were less than the sum of the figures in the row for that sector?

Calculate (a) \(P^{2}=P \cdot P\) (b) \(P^{4}=P^{2} \cdot P^{2}\) and \(\left(\right.\) c) \(P^{8} .\) Round all entries to four decimal places.) (d) Without computing it explicitly, find \(P^{1000}\). $$ P=\left[\begin{array}{lll} 0.25 & 0.25 & 0.50 \\ 0.25 & 0.25 & 0.50 \\ 0.25 & 0.25 & 0.50 \end{array}\right] $$

Find a scenario in which it would be useful to "multiply" two row vectors according to the rule $$ \left[\begin{array}{lll} a & b & c \end{array}\right]\left[\begin{array}{lll} d & e & f \end{array}\right]=\left[\begin{array}{lll} a d & b e & c f \end{array}\right] . $$
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