/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 Compute the products. Some of th... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 11

Compute the products. Some of these may be undefined.Exercises marked I should be done using technology. The others should be done two ways: by hand and by using technology where possible. \(\left[\begin{array}{llll}1 & -1 & 2 & 3\end{array}\right]\left[\begin{array}{rrr}-1 & 2 & 0 \\ 2 & -1 & 0 \\ 0 & 5 & 2 \\ -1 & 8 & 1\end{array}\right]\)

Short Answer

Expert verified
The product of the given matrices is: \(\left[\begin{array}{lll}-6 & 37 & 7\end{array}\right]\).

Step by step solution

01

Matrix dimensions and compatibility

First, we note down the dimensions of the given matrices and verify the compatibility for multiplication. Matrix A (1x4): \(\left[\begin{array}{llll}1 & -1 & 2 & 3\end{array}\right]\) Matrix B (4x3): \(\left[\begin{array}{rrr}-1 & 2 & 0 \\\ 2 & -1 & 0 \\\ 0 & 5 & 2 \\\ -1 & 8 & 1\end{array}\right]\) Since the number of columns of Matrix A (4) is equal to the number of rows of Matrix B (4), they are compatible for multiplication. The resulting matrix will have dimensions 1x3.
02

Multiply the matrices by hand

We will now multiply the matrices element by element. For the resulting matrix, C (1x3): C[1, 1] = A[1, 1] * B[1, 1] + A[1, 2] * B[2, 1] + A[1, 3] * B[3, 1] + A[1, 4] * B[4, 1] C[1, 1] = (1 * -1) + (-1 * 2) + (2 * 0) + (3 * -1) = -1 - 2 - 3 = -6 C[1, 2] = A[1, 1] * B[1, 2] + A[1, 2] * B[2, 2] + A[1, 3] * B[3, 2] + A[1, 4] * B[4, 2] C[1, 2] = (1 * 2) + (-1 * -1) + (2 * 5) + (3 * 8) = 2 + 1 + 10 + 24 = 37 C[1, 3] = A[1, 1] * B[1, 3] + A[1, 2] * B[2, 3] + A[1, 3] * B[3, 3] + A[1, 4] * B[4, 3] C[1, 3] = (1 * 0) + (-1 * 0) + (2 * 2) + (3 * 1) = 0 + 0 + 4 + 3 = 7 The resulting matrix C (1x3) is: \(\left[\begin{array}{lll}-6 & 37 & 7\end{array}\right]\)
03

Multiply the matrices using technology

We can also use technology (such as calculator, Python, or MATLAB) to multiply the given matrices. Here, we'll use Python code as an example: ```python import numpy as np A = np.array([[1, -1, 2, 3]]) B = np.array([[-1, 2, 0], [2, -1, 0], [0, 5, 2], [-1, 8, 1]]) C = np.dot(A, B) print(C) ``` The output will be: `[[-6, 37, 7]]` Both by hand and by using technology, we get the same resulting matrix: \(\left[\begin{array}{lll}-6 & 37 & 7\end{array}\right]\)

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Key Concepts

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

Matrix Dimensions
Understanding matrix dimensions is essential when dealing with matrix operations. A matrix is basically an array of numbers arranged in rows and columns. For instance, look at matrix A: \[\begin{array}{llll}1 & -1 & 2 & 3\end{array}\]It has 1 row and 4 columns, so it is a 1x4 matrix. Matrix B, on the other hand, looks like this:\[\begin{array}{rrr}-1 & 2 & 0 \2 & -1 & 0 \0 & 5 & 2 \-1 & 8 & 1\end{array}\]This matrix has 4 rows and 3 columns, thus it's a 4x3 matrix. Each dimension is a crucial piece of information that tells us not only the size of the matrix but also guides us when performing operations like multiplication.
Matrix Compatibility
Matrix multiplication is not as straightforward as multiplying individual numbers. To determine if two matrices are compatible for multiplication, you must compare their dimensions. Specifically, the number of columns in the first matrix should equal the number of rows in the second matrix.

In our exercise, matrix A is a 1x4 matrix and matrix B is a 4x3 matrix. Since the number of columns in matrix A (4) matches the number of rows in matrix B (4), these matrices are compatible, and multiplication is possible. When these matrices are multiplied, the resulting matrix will have dimensions equal to the number of rows of the first matrix and the number of columns of the second matrix. In this case, the product will be a 1x3 matrix.
  • The "inner" dimensions (4 in A, 4 in B) must match for multiplication to be feasible.
  • The "outer" dimensions (1 in A, 3 in B) define the size of the resulting matrix.
Using Technology for Calculations
Technology simplifies matrix multiplication, which can be otherwise tedious and error-prone if done manually, especially with large matrices. Various tools like calculators, Python, or MATLAB, can perform these computations efficiently. Let's focus on Python for this task:With Python's NumPy library, you can succinctly carry out matrix multiplication. Here鈥檚 a snippet explaining how:```pythonimport numpy as npA = np.array([[1, -1, 2, 3]])B = np.array([[-1, 2, 0], [2, -1, 0], [0, 5, 2], [-1, 8, 1]])C = np.dot(A, B)print(C)```The function `np.dot()` computes the matrix product, producing the result:\[\begin{array}{lll}-6 & 37 & 7\end{array}\]When utilizing software tools, ensure that your matrix data is correctly formatted and that you understand the functions used. This helps verify results against manual calculations and builds confidence in using technology for more complex problems.

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Most popular questions from this chapter

Compare addition and multiplication of \(1 \times 1\) matrices to the arithmetic of numbers.

Textbook Writing You are writing a college-level textbook on finite mathematics, and are trying to come up with the best combination of word problems. Over the years, you have accumulated a collection of amusing problems, serious applications, long complicated problems, and "generic" problems. \({ }^{25}\) Before your book is published, it must be scrutinized by several reviewers who, it seems, are never satisfied with the mix you use. You estimate that there are three kinds of reviewers: the "no-nonsense" types who prefer applications and generic problems, the "dead serious" types, who feel that a collegelevel text should be contain little or no humor and lots of long complicated problems, and the "laid-back" types, who believe that learning best takes place in a light-hearted atmosphere bordering on anarchy. You have drawn up the following chart, where the payoffs represent the reactions of reviewers on a scale of \(-10\) (ballistic) to \(+10\) (ecstatic): Reviewers ou \begin{tabular}{|l|c|c|c|} \hline & No-Nonsense & Dead Serious & Laid-Back \\ \hline Amusing & \(-5\) & \(-10\) & 10 \\ \hline Serious & 5 & 3 & 0 \\ \hline Long & \(-5\) & 5 & 3 \\ \hline Generic & 5 & 3 & \(-10\) \\ \hline \end{tabular} a. Your first draft of the book contained no generic problems, and equal numbers of the other categories. If half the reviewers of your book were "dead serious" and the rest were equally divided between the "no-nonsense" and "laid-back" types, what score would you expect? b. In your second draft of the book, you tried to balance the content by including some generic problems and eliminating several amusing ones, and wound up with a mix of which one eighth were amusing, one quarter were serious, three eighths were long, and a quarter were generic. What kind of reviewer would be least impressed by this mix? c. What kind of reviewer would be most impressed by the mix in your second draft?

What would it mean if the technology matrix \(A\) were the zero. matrix?

Is it possible for \(a 2 \times 3\) matrix to equal a \(3 \times 2\) matrix? Explain.

More Retail Discount Wars Your Abercrom B men's fashion outlet has a \(30 \%\) chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a \(70 \%\) chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a \(20 \%\) chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: Abercrom \(\mathbf{A}\) \(\begin{array}{ll}\text { Torn Shirts } & \text { Empty Shirts }\end{array}\) Mechanics \(\left[\begin{array}{rr}10 & -40 \\ -30 & 50\end{array}\right]\) Abercrom B \(\quad\) Military What is the expected resulting effect on your customer base?
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