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Chapter 3: Problem 44

Calculate (a) \(P^{2}=P \cdot P\) (b) \(P^{4}=P^{2} \cdot P^{2}\) and \(\left(\right.\) c) \(P^{8} .\) Round all entries to four decimal places.) (d) Without computing it explicitly, find \(P^{1000}\). $$ P=\left[\begin{array}{lll} 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \\ 0.3 & 0.3 & 0.4 \end{array}\right] $$

Short Answer

Expert verified
The calculations for the given powers of P are as follows: \(P^2 = \left[\begin{array}{lll} 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \end{array}\right]\) \(P^4 = \left[\begin{array}{lll} 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \end{array}\right]\) \(P^8 = \left[\begin{array}{lll} 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \end{array}\right]\) Since the matrix remains the same after each multiplication of P by itself, we conclude that \(P^{1000}\) is also equal to: \(P^{1000} = \left[\begin{array}{lll} 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \\ 0.3000 & 0.3000 & 0.4000 \end{array}\right]\)

Step by step solution

01

1. Calculate P^2 = P * P and round to four decimal places

To calculate the square of the matrix P, we need to perform matrix multiplication with itself: \(P^2 = P \cdot P\) We find the product by multiplying each element of the first row of P by its corresponding entry in the first column of P, and then add the products. The result goes in the first entry of \(P^2\). We repeat this process for all entries in the matrix. After calculating the values, we will round the entries to four decimal places.
02

2. Calculate P^4 = P^2 * P^2 and round to four decimal places

To calculate the fourth power of the matrix P, we need to perform matrix multiplication using the \(P^2\) result that we found in step 1: \(P^4 = P^2 \cdot P^2\) Similarly, we find the product by multiplying each element of the first row of \(P^2\) by its corresponding entry in the first column of \(P^2\), and then add the products. The result goes in the first entry of \(P^4\). We repeat this process for all entries in the matrix. After calculating the values, we will round the entries to four decimal places.
03

3. Calculate P^8 = P^4 * P^4 and round to four decimal places

To calculate the eighth power of the matrix P, we need to perform matrix multiplication using the \(P^4\) result that we found in step 2: \(P^8 = P^4 \cdot P^4\) Again, we find the product by multiplying each element of the first row of \(P^4\) with its corresponding entry in the first column of \(P^4\), and then add the products. The result goes in the first entry of \(P^8\). We repeat this process for all entries in the matrix. After calculating the values, we will round the entries to four decimal places.
04

4. Observe the pattern to find P^1000 without performing calculations

By observing the results for \(P^2\), \(P^4\), and \(P^8\), we may notice a pattern that emerges as we continually multiply P with itself. If the pattern is consistent, we can predict what \(P^{1000}\) would look like without explicitly computing it.

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Key Concepts

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

Matrix Multiplication
Understanding how to multiply matrices is pivotal in the study of linear algebra and its applications. Matrix multiplication isn't like your usual arithmetic; it's a bit more intricate. To multiply two matrices, let's call them A and B, you take the rows of A and columns of B and perform what is known as the dot product.

For instance, if you have a 3x3 matrix like the one shown in our exercise, you multiply the elements of the first row of A with the corresponding elements of the first column of B, add all these products together, and place the sum in the corresponding position in the resulting matrix, let's label it AB. Repeat this process for each row and column pair.

It's important to remember matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Also, the order matters - multiplying matrix A by matrix B will not necessarily give the same result as multiplying B by A.
Powers of a Matrix
When we talk about powers of a matrix, we're essentially referring to multiplying the matrix by itself a certain number of times. It鈥檚 similar to taking powers of numbers, but here, instead of simple multiplication, we're using matrix multiplication rules.

For example, the matrix P squared, denoted as P虏, means matrix P multiplied by itself once. A key aspect to note is that matrix powers only make sense for square matrices (those with the same number of rows and columns). Also, it's crucial that we maintain the sequence of multiplication because, unlike regular numbers, matrix multiplication is not commutative. With powers of matrices, we sometimes discover fascinating patterns as we increase the power, which can significantly simplify calculations, especially for higher powers like P鹿鈦扳伆鈦�.
Patterns in Matrix Powers
As we calculate higher powers of a matrix, a pattern may emerge in the elements of the resulting matrices. Spotting these patterns can save a lot of computational effort. In the provided exercise, after calculating the second, fourth, and eighth powers of matrix P, one could observe that the matrix begins to stabilize toward a certain consistent form.

This phenomenon occurs because of the unique properties inherent in the structure of P, and such behaviors are crucial in determining the long-term behavior of systems modeled by such matrices. Once the pattern is recognized, mathematicians can forecast the outcome of much higher powers without actual computation, as demonstrated in the exercise's step 4, which smartly anticipates what P鹿鈦扳伆鈦� would look like based on the established pattern.

Understanding this concept helps in various fields, including computer science, where matrix exponentiation is used in algorithms to speed up calculations for large powers.

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Most popular questions from this chapter

Calculate (a) \(P^{2}=P \cdot P\) (b) \(P^{4}=P^{2} \cdot P^{2}\) and \(\left(\right.\) c) \(P^{8} .\) Round all entries to four decimal places.) (d) Without computing it explicitly, find \(P^{1000}\). $$ P=\left[\begin{array}{lll} 0.25 & 0.25 & 0.50 \\ 0.25 & 0.25 & 0.50 \\ 0.25 & 0.25 & 0.50 \end{array}\right] $$

Translate the given matrix equations into svstems of linear equations. $$ \left[\begin{array}{rrr} 1 & -1 & 4 \\ -\frac{1}{3} & -3 & \frac{1}{3} \\ 3 & 0 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right]=\left[\begin{array}{r} -3 \\ -1 \\ 2 \end{array}\right] $$

(Staff Cutbacks Frank Tempest manages a large snowplow service in Manhattan, Kansas, and is alarmed by the recent weather trends; there have been no significant snowfalls since 1993\. He is therefore contemplating laying off some of his workers, but is unsure about whether to lay off 5,10 , or 15 of his 50 workers. Being very methodical, he estimates his annual net profits based on four possible annual snowfall figures: 0 inches, 20 inches, 40 inches and 60 inches. (He takes into account the fact that, if he is running a small operation in the face of a large annual snowfall, he will lose business to his competitors because he will be unable to discount on volume.) a. During the past 10 years, the region has had 0 inches twice, 20 inches twice, 40 inches three times, and 60 inches three times. Based on this information, how many workers should Tempest lay off, and how much would it cost him? b. There is a \(50 \%\) chance that Tempest will lay off 5 workers and a \(50 \%\) chance that he will lay off 15 workers. What is the worst thing Nature can do to him in terms of snowfall? How much would it cost him? c. The Gods of Chaos (who control the weather) know that Tempest is planning to use the strategy in part (a), and are determined to hurt Tempest as much as possible. Tempest, being somewhat paranoid, suspects it too. What should he do?

Revenue Recall the Left Coast Bookstore chain from the preceding section. In January, it sold 700 hardcover books, 1,300 softcover books, and 2,000 plastic books in San Francisco; it sold 400 hardcover, 300 softcover, and 500 plastic books in Los Angeles. Now, hardcover books sell for \(\$ 30\) each, softcover books sell for \(\$ 10\) each, and plastic books sell for \(\$ 15\) each. Write a column matrix with the price data and show how matrix multiplication (using the sales and price data matrices) may be used to compute the total revenue at the two stores.

A matrix is skew-symmetric or antisymmetric if it is equal to the negative of its transpose. Give an example of a. a nonzero \(2 \times 2\) skew-symmetric matrix and \(\mathbf{b}\). a nonzero \(3 \times 3\) skew-symmetric matrix.
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