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Chapter 8: Problem 2

A matrix is ________ if the number of rows equals the number of columns.

Short Answer

Expert verified
A matrix is square if the number of rows equals the number of columns.

Step by step solution

01

Understand the concept of a matrix

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Each individual entry within the matrix is often denoted by two indices, one for the row and one for the column.
02

Identify the characteristics of the given matrix

In this particular case, the number of rows is equal to the number of columns. This does not occur for every matrix, but is a unique characteristic of the matrix in question.
03

Name the type of matrix

When the number of rows equals the number of columns in a matrix, that matrix is known as a square matrix.

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Key Concepts

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

Matrix
A matrix is a fascinating mathematical concept that's used to organize data. It's essentially a rectangular array made up of numbers, symbols, or other values. These are arranged neatly into rows and columns. Think of a matrix like a table or a grid, one filled with various entries. Matrices are very useful when solving problems related to systems of equations, transformations, and in areas like computer graphics. In essence, a matrix is a handy tool that helps us organize and manipulate numbers or data efficiently.
Rows and Columns
Rows and columns are the structural backbone of a matrix, giving it shape and form. Rows run horizontally across the matrix, while columns extend vertically. Each cell within the matrix sits at the intersection of a particular row and column. Essentially, rows and columns allow us to navigate and reference specific entries in a matrix, ensuring we can find and use the information we need easy-peasy. Understanding how rows and columns work is crucial for effectively working with matrices.
This is especially important when you're trying to identify or define the size of a matrix. The number of rows and columns helps determine the matrix's dimensions.
Rectangular Array
A matrix is often described as a rectangular array, which essentially means it's laid out in a grid-like format. This rectangular setup is crucial because it sets matrices apart from other mathematical entities.
The regular arrangement of the rectangular array ensures that we can perform operations, like addition or multiplication, with a degree of order and precision. Matrices, as rectangular arrays, can be square if the number of rows matches the number of columns. However, they can also vary in shape with different numbers of rows and columns. This versatile layout enables us to use matrices in many different applications, from computer science to physics.
Indices in a Matrix
When digging into matrices, understanding indices is vital. Indices are numbers that help us to precisely locate elements within a matrix. Every element in a matrix has a unique position, defined by its row and column indices.
Typically, indices are written in pairs. The row index comes first, followed by the column index, like this: \(a_{ij}\), where \(i\) represents the row number and \(j\) represents the column number. These indices are key to accessing any part of a matrix efficiently and enable us to perform efficient calculations or transformations, focusing exactly where adjustments need to be made.

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Most popular questions from this chapter

In Exercises 21-32, use a determinant and the given vertices of a triangle to find the area of the triangle. \((-4, 2)\), \((0, \frac{7}{2})\), \((3, -\frac{1}{2})\)

In Exercises 63-70, find (a) \(|A|\), (b) \(|B|\), (c) \(AB\), and (d) \(|AB|\). \(A = \left[ \begin{array}{r} -1 && 2 && 1 \\ 1 && 0 && 1 \\ 0 && 1 && 0 \end{array} \right]\), \(B = \left[ \begin{array}{r} -1 && 0 && 0 \\ 0 && 2 && 0 \\ 0 && 0 && 3 \end{array} \right]\)

In Exercises 21-32, use a determinant and the given vertices of a triangle to find the area of the triangle. \((-2, 4)\), \((1, 5)\), \((3, -2)\)

In Exercises 77-84, solve for \(x\). \(\left| \begin{array}{c} x+4 & -2 \\ 7 & x-5 \end{array} \right| = 0\)

PROPERTIES OF DETERMINANTS In Exercises 97-99, a property of determinants is given (\(A\) and \(B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by interchanging two rows of or interchanging two columns of \(A\), then \(|B| = -|A|\). (a) \(\left| \begin{array}{r} 1 && 3 & 4 \\ -7 && 2 & -5 \\ 6 && 1 & 2 \end{array} \right| = -\left| \begin{array}{r} 1 & 4 && 3 \\ -7 & -5 && 2 \\\ 6 & 2 && 1 \end{array} \right|\) (b) \(\left| \begin{array}{r} 1 && 3 && 4 \\ -2 && 2 && 0 \\ 1 && 6 && 2 \end{array} \right| = -\left| \begin{array}{r} 1 && 6 && 2 \\ -2 && 2 && 0 \\\ 1 && 3 && 4 \end{array} \right|\)
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