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Chapter 9: Problem 18

Concept Check Find the dimension of each matrix. Identify any square, column, or row matrices. $$[-9]$$

Short Answer

Expert verified
The matrix \([-9]\) has dimensions 1x1 and is a square, row, and column matrix.

Step by step solution

01

Identify the Matrix Size

First, determine the dimensions of the matrix. A matrix's dimensions are given by the number of rows and columns it contains. This matrix \([-9]\) consists of 1 row and 1 column.
02

Check for Special Types of Matrix

A matrix is classified as a square matrix if it has an equal number of rows and columns. A column matrix has a single column, while a row matrix has a single row.
03

Determine the Type

Given that the matrix \([-9]\) has the same number of rows and columns (1 row and 1 column), it is a square matrix. Since it also has just one row and one column, it is both a row matrix and a column matrix.

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Key Concepts

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

Understanding Square Matrices
A square matrix is a special type of matrix that has the same number of rows and columns. In other words, for a matrix to be classified as a square matrix, its dimensions must follow the form \( n \times n \), where \( n \) is a positive integer. Square matrices are important in various mathematical fields, including linear algebra, as they have unique properties.
Common examples of square matrices include:
  • The identity matrix: an \( n \times n \) matrix with ones on the diagonal and zeros elsewhere.
  • The zero matrix: an \( n \times n \) matrix with all entries being zero.

From the exercise, we see that the matrix \([-9]\) is a \(1 \times 1\) matrix, making it a square matrix. This unique matrix has only one element, positioned at the intersection of the first row and first column.
Understanding Row Matrices
A row matrix, also known as a row vector, is a matrix that has exactly one row and one or more columns. The general form of a row matrix is \(1 \times n\), where \( n \) represents the number of columns. Row matrices are commonly used in different areas of mathematics and data science to represent vectors in a simplified form.
Key characteristics of row matrices include:
  • They have only one row.
  • They contain multiple columns.
  • They are often used to represent solutions to equations or sets of data.

From the given exercise, the matrix \([-9]\) is a \(1 \times 1\) matrix. Despite being a square matrix, it also fits the criteria of a row matrix since it has exactly one row.
Understanding Column Matrices
A column matrix, or column vector, is a matrix that consists of a single column and one or more rows. The general form of a column matrix is \(m \times 1\), where \( m \) represents the number of rows. Column matrices are widely used to represent vectors and are pivotal in linear algebra and vector calculus.
Characteristics of column matrices include:
  • They have only one column.
  • They contain multiple rows.
  • They are often used in transformations and linear mappings.

In the original exercise, the given matrix \([-9]\) is also a column matrix because it possesses a single column alongside its single row. Therefore, it simultaneously meets the criteria for being a square, row, and column matrix. This overlap of classifications is due to the unique \(1 \times 1\) dimension.

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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) Suppose the demand and price for a certain model of electric can opener are related by \(p=16-\frac{5}{4} q\), where \(p\) is price, in dollars, and \(q\) is demand, in appropriate units. Find the price when the demand is at each level. (a) 0 units (b) 4 units (c) 8 units

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{aligned} -2 x-2 y+3 z &=4 \\ 5 x+7 y-z &=2 \\ 2 x+2 y-3 z &=-4 \end{aligned}$$

Use a system of equations to solve each problem. Find the equation of the line \(y=a x+b\) that passes through the points \((3,-4)\) and \((-1,4)\)

Solve each problem. Yogurt sells three types of yogurt: nonfat, regular, and super creamy, at three locations. Location I sells 50 gal of nonfat, 100 gal of regular, and 30 gal of super creamy each day. Location II sells 10 gal of nonfat, and Location III sells 60 gal of nonfat each day. Daily sales of regular yogurt are 90 gal at Location II and 120 gal at Location III. At Location II, 50 gal of super creamy are sold each day, and 40 gal of super creamy are sold each day at Location III. (a) Write a \(3 \times 3\) matrix that shows the sales figures for the three locations, with the rows representing the three locations. (b) The incomes per gallon for nonfat, regular, and super creamy are \(\$ 12, \$ 10,\) and \(\$ 15,\) respectively. Write a \(1 \times 3\) or \(3 \times 1\) matrix displaying the incomes. (c) Find a matrix product that gives the daily income at each of the three locations. (d) What is Yagel's Yogurt's total daily income from the three locations?

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}$$
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