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Chapter 9: Problem 15

Concept Check Find the dimension of each matrix. Identify any square, column, or row matrices. $$\left[\begin{array}{rrrr} -6 & 8 & 0 & 0 \\ 4 & 1 & 9 & 2 \\ 3 & -5 & 7 & 1 \end{array}\right]$$

Short Answer

Expert verified
The matrix is 3x4 and is neither a square, row, nor column matrix.

Step by step solution

01

- Identify the number of rows

Count the number of horizontal lines of elements. There are 3 horizontal lines, hence 3 rows.
02

- Identify the number of columns

Count the number of vertical lines of elements. There are 4 vertical lines, hence 4 columns.
03

- Determine the dimension

Combine the number of rows and columns. The dimension of the matrix is 3x4 (3 rows and 4 columns).
04

- Identify if the matrix is square

A square matrix has the same number of rows and columns. Since this matrix has 3 rows and 4 columns, it is not square.
05

- Identify if the matrix is a row matrix

A row matrix has only one row. This matrix has 3 rows, so it is not a row matrix.
06

- Identify if the matrix is a column matrix

A column matrix has only one column. This matrix has 4 columns, so it is not a column matrix.

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Key Concepts

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

Row Matrix
A row matrix is a special type of matrix that has only one row. This means all the elements are lined up horizontally in a single row.
For example, the matrix \(\begin{array}{cccc} 4 & 5 & 6 & 7 \end{array}\) is a row matrix.
It has just one row and four different elements.
You will also notice it does not have multiple horizontal lines, making it distinct.
Row matrices are useful in various mathematical operations. They can represent coefficients in linear equations or be used in computer graphics.
To recognize a row matrix, simply check the number of horizontal lines. If there's only one, it鈥檚 a row matrix.
In our original exercise, the matrix with 3 rows and 4 columns is not a row matrix.
This is because it has more than one row.
Column Matrix
A column matrix, as the name suggests, is a matrix that has only one column.
This means all elements are arranged vertically in a single column.
For instance, let's look at \(\begin{array}{c} 3 \ 7 \ 9 \end{array}\).
This is a column matrix with three elements stacked one above the other.
Column matrices are pivotal in various fields such as statistics, physics, and economics.
They often represent vectors, which are crucial in understanding many concepts in these disciplines.
When trying to identify a column matrix, simply check the number of vertical lines. If there's only one, that's your column matrix.
Referring to our original exercise, the matrix with 3 rows and 4 columns is not a column matrix.
This is because it has more than one column.
Square Matrix
A square matrix is a type of matrix that has an equal number of rows and columns.
For example, \(\begin{array}{ccc} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{array}\) is a square matrix because it has 3 rows and 3 columns.
This property makes square matrices very special and they appear frequently in various branches of mathematics and engineering.
They are essential in operations such as finding determinants, eigenvalues, and solving linear equations.
To identify a square matrix, count the rows and columns. If the numbers are the same, that鈥檚 a square matrix.
Returning to our original exercise, the matrix provided is not a square matrix.
With 3 rows and 4 columns, the row and column count do not match.

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Most popular questions from this chapter

Solve each linear programming problem. Aid to Disaster Victims Earthquake victims in Haiti need medical supplies and bottled water. Each medical kit measures \(1 \mathrm{ft}^{3}\) and weighs 10 lb. Each container of water is also \(1 \mathrm{ft}^{3}\) but weighs \(20 \mathrm{lb}\). The plane can carry only \(80,000\) Ib with a total volume of \(6000 \mathrm{ft}^{3}\) Each medical kit will aid 4 people, and each container of water will serve 10 people. (a) How many of each should be sent to maximize the number of people helped? (b) If each medical kit could aid 6 people instead of 4, how would the results from part (a) change? PICTURE CANT COPY

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix \(A\) Eggs lour Sugar Shortening Milk \(\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A\) The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix \(B\) Large Lot Small Lot \(\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B\) (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a \(1 \times 4\) matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned}&y \geq|x+2|\\\&y \leq 6\end{aligned}$$

Gasoline Revenues The manufacturing process requires that oil refineries manufacture at least 2 gal of gasoline for each gallon of fuel oil. To meet the winter demand for fuel oil, at least 3 million gal per day must be produced. The demand for gasoline is no more than 6.4 million gal per day. If the price of gasoline is \(\$ 2.90\) per gal and the price of fuel oil is \(\$ 2.50\) per gal, how much of each should be produced to maximize revenue?

Solve each problem. Plate-Glass Sales The amount of plate-glass sales \(S\) (in millions of dollars) can be affected by the number of new building contracts \(B\) issued (in millions) and automobiles \(A\) produced (in millions). A plate-glass company in California wants to forecast future sales by using the past three years of sales. The totals for the three years are given in the table. To describe the relationship among these variables, we can use the equation $$ S=a+b A+c B $$. where the coefficients \(a, b,\) and \(c\) are constants that must be determined before the equation can be used. (Source: Makridakis, S., and S. Wheelwright, Forecasting Methods for Management, John Wiley and Sons.) (a) Substitute the values for \(S, A,\) and \(B\) for each year from the table into the equation \(S=a+b A+c B,\) and obtain three linear equations involving \(a, b,\) and \(c\) (b) Use a graphing calculator to solve this linear system for \(a, b,\) and \(c .\) Use matrix inverse methods. (c) Write the equation for \(S\) using these values for the coefficients. $$\begin{array}{|c|c|c|} \hline S & A & B \\ \hline 602.7 & 5.543 & 37.14 \\\ \hline 656.7 & 6.933 & 41.30 \\ \hline 778.5 & 7.638 & 45.62 \\ \hline \end{array}$$ (d) For the next year it is estimated that \(A=7.752\) and \(B=47.38 .\) Predict \(S .\) (The actual value for \(S\) was \(877.6 .\) ) (e) It is predicted that in 6 yr, \(A=8.9\) and \(B=66.25 .\) Find the value of \(S\) in this situation and discuss its validity.
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