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Chapter 3: Problem 77

Explain the difference between a matrix and a determinant. Give an example of each.

Short Answer

Expert verified
A matrix is an array of numbers, whereas a determinant is a scalar computed from a square matrix. Example: \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\) is a matrix; \(det(B) = -2\) for \(B = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix}\).

Step by step solution

01

Define a Matrix

A matrix is a rectangular array of numbers arranged in rows and columns. Each element of the matrix is identified by its row number and column number. For example, a 2x3 matrix (2 rows and 3 columns) could look like this:\[A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}\]
02

Define a Determinant

The determinant is a scalar value that can be computed from a square matrix (a matrix with the same number of rows and columns). It provides important properties of the matrix, like whether it is invertible. The determinant of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as:\[det = ad - bc\]
03

Example of a Matrix

Consider the matrix \(B\) given by:\[B = \begin{bmatrix} 7 & 8 \ 9 & 10 \end{bmatrix}\]This matrix has 2 rows and 2 columns, which makes it square.
04

Calculate the Determinant of the Example Matrix

For the matrix \(B\), the determinant is calculated using the formula for a 2x2 matrix:\[det(B) = (7)(10) - (8)(9) = 70 - 72 = -2\]So the determinant of matrix \(B\) is \(-2\).

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Key Concepts

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

Determinant
A determinant is a special number associated with square matrices. This number gives us valuable insight into the matrix's properties. You can think of a determinant as a single value that condenses information about the matrix.

To calculate the determinant of a 2x2 matrix, say \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), use this formula:
  • Multiply the top left element \(a\) by the bottom right element \(d\).
  • Multiply the top right element \(b\) by the bottom left element \(c\).
  • Subtract the product of the second step from the first: \(det = ad - bc\).
This calculated number determines several attributes of the matrix, such as whether the matrix is invertible, zero or not. If the determinant is zero, the matrix cannot be inverted.
Invertible Matrix
An invertible matrix, also known as a non-singular or non-degenerate matrix, is a type of square matrix that has an inverse. The inverse of a matrix is another matrix that, when multiplied with the original matrix, yields the identity matrix.

For a 2x2 matrix to be invertible, its determinant must be non-zero. This requirement ensures there's no mathematical conflict while performing matrix inversion operations.
Here's how it works:
  • Consider a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
  • The inverse of this matrix exists only if the determinant \( det = ad - bc \) is not zero.
  • If an inverse exists, it can be calculated, allowing you to perform operations to "undo" the effects of the original matrix.
This quality is critical in various fields like computer graphics and solving systems of linear equations.
2x2 Matrix
A 2x2 matrix is the simplest form of a square matrix with two rows and two columns. Each element of the matrix can be manipulated for various operations, such as addition, subtraction, multiplication, or finding the determinant.

The general representation of a 2x2 matrix is:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
Such matrices appear frequently in linear algebra because they are easy to understand and serve as building blocks for more complex structures. In real-world applications, 2x2 matrices can model simple transformations and basic operations. For example, they can represent scaling and rotational actions in graphical applications.
Square Matrix
A square matrix is defined as a matrix with the same number of rows and columns. Common examples include 2x2, 3x3, and so on. Square matrices are important because many operations in linear algebra, like finding the determinant or inverse, are only defined for them.

Key properties of square matrices include:
  • Symmetric: A square matrix is symmetric if it's identical to its transpose (its rows and columns are swapped).
  • Diagonal: Only the elements on the main diagonal, which runs from the top left to the bottom right, may be non-zero.
  • Identity: A special kind of square matrix with "1"s on the main diagonal and "0"s elsewhere. Multiplying any matrix by an identity matrix does not change the original matrix.
These characteristics make square matrices a central topic of study, especially in solving linear systems and in the field of mathematical transformations.

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Most popular questions from this chapter

Use matrices to solve each system of equations. If the equations of a system are dependent or if a system is inconsistent, state this. $$ \left\\{\begin{array}{l} 5 x+3 y=4 \\ 3 y-4 z=4 \\ x+z=1 \end{array}\right. $$

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} x+y=1 \\ \frac{1}{2} y+z=\frac{5}{2} \\ x-z=-3 \end{array}\right. $$

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 3 x-16=5 y \\ -3 x+5 y-33=0 \end{array}\right. $$

Use Cramer's rule to solve each system of equations. If a system is inconsistent or if the equations are dependent, so indicate. $$ \left\\{\begin{array}{l} 4 x-3 y=1 \\ 6 x-8 z=1 \\ 2 y-4 z=0 \end{array}\right. $$

Fill in the blanks. The _______ of a three-dimensional geometric solid is the amount of space it encloses.
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