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Chapter 9: Problem 2

State the dimension of the matrix. $$\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \\ 0 & 2 & 11 & 3 \end{array}\right]$$

Short Answer

Expert verified
The dimension of the matrix is \(2 \times 4\).

Step by step solution

01

Identify Rows

To determine the dimension of a matrix, we first need to count the number of rows. Rows are the horizontal lines of entries in the matrix. In this matrix, there are 2 rows: \([-1, 5, 4, 0]\) and \([0, 2, 11, 3]\).
02

Identify Columns

Next, we count the number of columns. Columns are the vertical lines of entries in the matrix. There are 4 columns: 1. \([-1, 0]\)2. \([5, 2]\)3. \([4, 11]\)4. \([0, 3]\)
03

State the Dimension

The dimension of a matrix is given by the number of rows by the number of columns, written as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns. For this matrix, it has 2 rows and 4 columns, therefore, the dimension is \(2 \times 4\).

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Key Concepts

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

Understanding Rows and Columns in a Matrix
In a matrix, **rows** and **columns** are fundamental elements that define its structure. Rows are horizontal lines that contain elements arranged side by side. Similarly, columns are vertical lines of elements stacked on top of one another. Think of rows as the horizontal groupings, and columns as the vertical groupings.
When working with matrices, it is essential to understand these concepts to interpret and manipulate them correctly. This matrix is laid out in a rectangular shape:
  • The rows are arranged horizontally, such as \([ -1, 5, 4, 0 ]\) and \([ 0, 2, 11, 3 ]\).
  • The columns are organized vertically, one above the other, like \([-1, 0]\), \([5, 2]\), \([4, 11]\), and \([0, 3]\).
Understanding how to identify rows and columns is the first step in determining the dimensions of a matrix.
Exploring the Concept of a Matrix
A **matrix** is a rectangular array of numbers or expressions arranged in rows and columns. It is a fundamental concept in various fields, such as mathematics, physics, and computer science. Each element in a matrix is identified by its position, given as \(a_{ij}\), where \(i\) is the row number, and \(j\) is the column number.
Matrices can vary in size, and their dimensions are noted as \(m \times n\), where \(m\) refers to the number of rows and \(n\) the number of columns.
Matrices are useful in operations such as addition, subtraction, and multiplication. However, operations depend on the matrix dimensions, so understanding the size and structure of a matrix is crucial. For example, two matrices can only be added if they have the same dimensions.
Importance of Matrices in Precalculus
**Matrices in precalculus** play a vital role in solving various problems involving systems of equations and transformations. They offer a structured way to handle multiple linear equations simultaneously, which is common in precalculus.
Matrices help simplify complex problems into manageable calculations, making it easier to see relationships between different variables.
  • They are used to represent systems of linear equations. Each row can correspond to an equation.
  • Matrices can facilitate operations like transformations in geometry, providing a way to perform rotations, translations, and reflections efficiently.
In essence, mastering the concept of matrices in precalculus is essential for ensuring a smooth transition to more advanced topics, like linear algebra and calculus.

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Most popular questions from this chapter

Express the function (or rule) in words. $$k(x)=\sqrt{x+2}$$

A biologist is performing an experiment on the effects of various combinations of vitamins. She wishes to feed each of her laboratory rabbits a diet that contains exactly \(9 \mathrm{mg}\) of niacin, \(14 \mathrm{mg}\) of thiamin, and \(32 \mathrm{mg}\) of riboflavin. She has available three different types of commercial rabbit pellets; their vitamin content (per ounce) is given in the table. How many ounces of each type of food should each rabbit be given daily to satisfy the experiment requirements? $$\begin{array}{|l|c|c|c|} \hline & \text { Type A } & \text { Type B } & \text { Type C } \\ \hline \text { Niacin (mg) } & 2 & 3 & 1 \\ \text { Thiamin (mg) } & 3 & 1 & 3 \\ \text { Riboflavin (mg) } & 8 & 5 & 7 \\ \hline \end{array}$$

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Find the inverse of the matrix. For what value(s) of \(x\) if any, does the matrix have no inverse? $$\left[\begin{array}{rrr} 1 & e^{x} & 0 \\ e^{x} & -e^{2 x} & 0 \\ 0 & 0 & 2 \end{array}\right]$$

Nutrition A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce: \begin{array}{|l|ccc|} \hline & \text { Type A } & \text { Type B } & \text { Type C } \\ \hline \text { Folic acid (mg) } & 3 & 1 & 3 \\ \text { Choline (mg) } & 4 & 2 & 4 \\ \text { Inositol (mg) } & 3 & 2 & 4 \\ \hline \end{array} (a) Find the inverse of the matrix $$\left[\begin{array}{lll} 3 & 1 & 3 \\ 4 & 2 & 4 \\ 3 & 2 & 4 \end{array}\right]$$ and use it to solve the remaining parts of this problem. (b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain \(10 \mathrm{mg}\) of folic acid, \(14 \mathrm{mg}\) of choline, and \(13 \mathrm{mg}\) of inositol? (c) How much of each food is needed to supply \(9 \mathrm{mg}\) of folic acid, 12 mg of choline, and 10 mg of inositol? (d) Will any combination of these foods supply \(2 \mathrm{mg}\) of folic acid, \(4 \mathrm{mg}\) of choline, and \(11 \mathrm{mg}\) of inositol?
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