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Chapter 6: Problem 5

Find the dimension of each matrix. Identify any square, column, or row matrices. Do not use a calculator. $$\left[\begin{array}{l} 2 \\ 4 \end{array}\right]$$

Short Answer

Expert verified
The matrix has dimensions 2x1 and is a column matrix.

Step by step solution

01

Determine Dimensions

To find the dimensions of the matrix, count the number of rows and the number of columns. The given matrix is \[\begin{bmatrix}2 \4\end{bmatrix}\]. This matrix has 2 rows and 1 column.
02

Identify Type of Matrix

Now, classify the type of matrix you have: - Row matrix: A matrix with exactly one row. - Column matrix: A matrix with exactly one column. - Square matrix: A matrix with an equal number of rows and columns. The given matrix has 2 rows and 1 column, which means it is a column matrix. It is not a square matrix since the number of rows and columns are not equal.
03

Conclusion

The given matrix has dimensions of 2x1, indicating it has 2 rows and 1 column. It is classified as a column matrix.

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Key Concepts

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

Types of Matrices
Matrices come in different forms based on their shape and number of rows and columns. Understanding the various types of matrices can help identify their specific properties and uses. Here are the main types you will encounter in mathematics:
  • Row Matrix: This matrix has only one row but can have multiple columns. For example, a matrix with dimensions 1x3 is a row matrix.
  • Column Matrix: As the name suggests, this type has a single column but can contain several rows. An example is a matrix with dimensions 5x1.
  • Square Matrix: In this matrix, the number of rows equals the number of columns, like a 3x3 or 4x4 matrix. This equal structure makes square matrices significant in various calculations, such as determinants and inverses.

In addition to these, other specialized matrices include identity matrices and zero matrices, each with unique characteristics. Recognizing these types can simplify many linear algebra tasks.
Column Matrix
A column matrix, sometimes called a column vector, is one of the simpler forms of matrices to identify. Its key feature is having only one column. Despite being simple, column matrices are very useful and play a crucial role in many areas of mathematics and applied sciences.

Properties of Column Matrices

  • Only one column, but can have several rows. These rows provide the dimensions of the matrix.
  • The dimensions of a column matrix are represented as, for example, 3x1, which means 3 rows and 1 column.

Column matrices often represent vectors in mathematical computations, especially in physics and engineering. They are also used in operations like matrix multiplication, where you might see them interact with other matrices to transform data or solve systems of equations. Understanding how to manipulate column matrices will strengthen your grasp of vector and matrix operations.
Square Matrix
A square matrix distinguishes itself by having the same number of rows as columns, forming an equal-sided shape. This unique quality makes them pivotal in many mathematical operations.

Features and Importance

  • A square matrix can be of dimension NxN, such as 2x2, 3x3, or NxN.
  • The main diagonal of a square matrix, which runs from the top left to the bottom right, has special significance in determinant calculations and trace operations.

Square matrices are vital because they support operations that are unavailable to non-square matrices, such as calculating determinants, finding inverses, and eigenvalues. These features are key in systems of linear equations, transformations in geometry, and are the foundation of more complex mathematical theories. Mastering square matrices opens doors to a deeper understanding of mathematical concepts and applications.

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Most popular questions from this chapter

The break-even point for a company is the point where costs equal revenues. If both cost and revenue are expressed as linear equations, the break-even point is the solution of a linear system. In each exercise, \(C\) represents cost in dollars to produce x items, and R represents revenue in dollars from the sale of \(x\) items. Use the substitution method to find the break-even point in each case-that is, the point where \(C=R .\) Then find the value of \(C\) and \(R\) at that point. $$\begin{aligned}&C=4 x+125\\\&R=9 x-200\end{aligned}$$

Fawn Population To model spring fawn count \(F\) from adult antelope population \(A\), precipitation \(P\), and severity of winter \(W,\) environmentalists have used the equation. $$F=a+b A+c P+d W$$ where the coefficients \(a, b, c,\) and \(d\) are constants that must be determined before using the equation. The table lists the results of four different (representative) years. $$\begin{array}{|c|c|c|c|} \text { Fawns } & \text { Adults } & \begin{array}{c} \text { Precip. } \\ \text { (in inches) } \end{array} & \begin{array}{c} \text { Winter } \\ \text { Severity } \end{array} \\ \hline 239 & 871 & 11.5 & 3 \\ 234 & 847 & 12.2 & 2 \\ 192 & 685 & 10.6 & 5 \\ 343 & 969 & 14.2 & 1 \end{array}$$ A. Substitute the values for \(F, A, P,\) and \(W\) from the table for each of the four years into the given equation \(F=a+b A+c P+d W\) to obtain four linear equations involving \(a, b, c,\) and \(d\) B. Write a \(4 \times 5\) augmented matrix representing the system, and solve for \(a, b, c,\) and \(d\) C. Write the equation for \(F\), using the values from part (b) for the coefficients. D. If a winter has severity \(3,\) adult antelope population 960 and precipitation 12.6 inches, predict the spring fawn count. (Compare this with the actual count of \(320 .\) )

Given a square matrix \(A^{-1}\), find matrix \(A\). Let \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & -1\end{array}\right] .\) Show that \(A^{3}=I_{3},\) and use this result to find the inverse of \(A\).

In certain parts of the Rocky Mountains, deer are the main food source for mountain lions. When the deer population \(d\) is large, the mountain lions ( \(m\) ) thrive. However, a large mountain lion population drives down the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$\left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{cc} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 year? 2 years? (c) Consider part (b), but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of \(1 \$ 6\).

Given a square matrix \(A^{-1}\), find matrix \(A\). $$A^{-1}=\left[\begin{array}{rrr} \frac{2}{3} & -\frac{1}{3} & 0 \\ \frac{1}{3} & -\frac{5}{3} & 1 \\ \frac{1}{3} & \frac{1}{3} & 0 \end{array}\right]$$
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