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Chapter 2: Problem 24

Compute the indicated products. $$ 2\left[\begin{array}{rrr} 3 & 2 & -1 \\ 0 & 1 & 3 \\ 2 & 0 & 3 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{rrr} 1 & 2 & 0 \\ 0 & -1 & -2 \\ 1 & 3 & 1 \end{array}\right] $$

Short Answer

Expert verified
The short answer to the problem is: $$ \left[\begin{array}{rrr} 4 & 16 & 3 \\ -4 & -10 & 3 \\ 10 & 26 & 6 \end{array}\right] $$

Step by step solution

01

Compute the product of the matrices A and B

First, multiply the scalar 2 with the first matrix, then find the product of the two matrices: $$ A = 2\left[\begin{array}{rrr} 3 & 2 & -1 \\ 0 & 1 & 3 \\ 2 & 0 & 3 \end{array}\right] $$ $$ B = \left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ To multiply two matrices, we have to multiply each element of a row in the first matrix with the corresponding element of a column in the second matrix and sum the results. Perform the matrix product: $$ A\cdot B = \left[\begin{array}{rrr} 6 & 2 & -2 \\ 0 & 2 & 6 \\ 4 & 0 & 6 \end{array}\right] $$
02

Compute the product of the result with the third matrix

Now we will find the product of the result with the third matrix: $$ C = \left[\begin{array}{rrr} 1 & 2 & 0 \\ 0 & -1 & -2 \\ 1 & 3 & 1 \end{array}\right] $$ Perform the matrix product: $$ (A\cdot B)\cdot C = \left[\begin{array}{ccc} 6+0-2 & 12-2+6 & -2+4+1 \\ 0+0-4 & 0+2-12 & 0+2+1 \\ 4+0+6 & 8+0+18 & 0+0+6 \end{array}\right] = \left[\begin{array}{rrr} 4 & 16 & 3 \\ -4 & -10 & 3 \\ 10 & 26 & 6 \end{array}\right] $$ The product of the three matrices is: $$ \left[\begin{array}{rrr} 4 & 16 & 3 \\ -4 & -10 & 3 \\ 10 & 26 & 6 \end{array}\right] $$

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Key Concepts

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

Scalar Multiplication in Matrices
Scalar multiplication in matrices is a simple yet powerful operation in linear algebra. It involves multiplying every element of a matrix by a scalar (a single number), which in this case is 2.
Imagine you have a matrix like this:
  • \[\left[\begin{array}{rrr}3 & 2 & -1 \0 & 1 & 3 \2 & 0 & 3 \\end{array}\right]\]
When you perform scalar multiplication, you multiply the scalar value (2) with each element in the matrix to produce a new matrix. The matrix changes to:
  • \[\left[\begin{array}{rrr}6 & 4 & -2 \0 & 2 & 6 \4 & 0 & 6 \\end{array}\right]\]
This operation is very straightforward and helps scale all the values in the matrix uniformly.
It is useful in various mathematical computations, including transformations and linear equations.
Identity Matrix
The identity matrix plays a pivotal role in matrix multiplication. It is essentially the matrix equivalent of the number 1 in arithmetic.
Consider an identity matrix like this:
  • \[\left[\begin{array}{lll}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \\end{array}\right]\]
When you multiply any square matrix by an identity matrix of the same order, you get the original matrix back. This behavior makes it a neutral element in matrix multiplication.
For example, multiplying the matrix:
  • \[\left[\begin{array}{rrr}6 & 4 & -2 \0 & 2 & 6 \4 & 0 & 6 \\end{array}\right]\]
by the identity matrix results in the same:
  • \[\left[\begin{array}{rrr}6 & 4 & -2 \0 & 2 & 6 \4 & 0 & 6 \\end{array}\right]\]
Understanding the identity matrix is vital for processes that require maintaining or verifying the integrity of matrices during calculations.
Step-by-Step Matrix Computation
Walking through matrix computations step-by-step ensures clarity and prevents mistakes. Let's examine this through the product of three matrices:
First, start by applying scalar multiplication to matrix A, then multiply with the identity matrix B.
  • Scalar multiplication yields:\[\left[\begin{array}{rrr}6 & 4 & -2 \0 & 2 & 6 \4 & 0 & 6 \\end{array}\right]\]
  • Next, multiply with B, the identity matrix, and result remains the same.
  • Finally, compute the product with matrix C:\[\left[\begin{array}{rrr}1 & 2 & 0 \0 & -1 & -2 \1 & 3 & 1 \\end{array}\right]\]
Multiply each element of the rows from the resulting matrix with the corresponding columns of C, and add the results. For the first element in the final matrix:
  • \(6 \times 1 + 4 \times 0 - 2 \times 1 = 4\)
Continue this method to find all elements of the final product, resulting in:
  • \[\left[\begin{array}{rrr}4 & 16 & 3 \-4 & -10 & 3 \10 & 26 & 6 \\end{array}\right]\]
This comprehensive approach highlights the efficiency and accuracy gained through stepwise calculations.

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Most popular questions from this chapter

Mr. and Mrs. Garcia have a total of $$\$ 100,000$$ to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \% /\) year, while the bonds and the money market account pay \(8 \% /\) year and \(4 \%\) year, respectively. The Garcias have stipulated that the amount invested in the money market account should be equal to the sum of \(20 \%\) of the amount invested in stocks and \(10 \%\) of the amount invested in bonds. How should the Garcias allocate their resources if they require an annual income of $$\$$ 10,000$ from their investments?

Kaitlin and her friend Emma returned to the United States from a tour of four cities: Oslo, Stockholm, Copenhagen, and Saint Petersburg. They now wish to exchange the various foreign currencies that they have accumulated for U.S. dollars. Kaitlin has 82 Norwegian krones, 68 Swedish krones, 62 Danish krones, and 1200 Russian rubles. Emma has 64 Norwegian krones, 74 Swedish krones, 44 Danish krones, and 1600 Russian rubles. Suppose the exchange rates are U.S. \(\$ 0.1651\) for one Norwegian krone, U.S. \$0.1462 for one Swedish krone, U.S. \$0.1811 for one Danish krone, and U.S. \(\$ 0.0387\) for one Russian ruble. a. Write a \(2 \times 4\) matrix \(A\) giving the values of the various foreign currencies held by Kaitlin and Emma. (Note: The answer is not unique.) b. Write a column matrix \(B\) giving the exchange rate for the various currencies. c. If both Kaitlin and Emma exchange all their foreign currencies for U.S. dollars, how many dollars will each have?

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. A dietitian wishes to plan a meal around three foods. The percent of the daily requirements of proteins, carbohydrates, and iron contained in each ounce of the three foods is summarized in the following table: $$\begin{array}{lccc} \hline & \text { Food I } & \text { Food II } & \text { Food III } \\ \hline \text { Proteins }(\%) & 10 & 6 & 8 \\ \hline \text { Carbohydrates }(\%) & 10 & 12 & 6 \\ \hline \text { Iron }(\%) & 5 & 4 & 12 \\ \hline \end{array}$$ Determine how many ounces of each food the dietitian should include in the meal to meet exactly the daily requirement of proteins, carbohydrates, and iron \((100 \%\) of each).

Find the matrix \(A\) such that $$ \begin{array}{r} A\left[\begin{array}{rr} 1 & 0 \\ -1 & 3 \end{array}\right]=\left[\begin{array}{rr} -1 & -3 \\ 3 & 6 \end{array}\right] \\ \text { Hint: Let } A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] . \end{array} $$

Ethan just returned to the United States from a Southeast Asian trip and wishes to exchange the various foreign currencies that he has accumulated for U.S. dollars. He has 1200 Thai bahts, 80,000 Indonesian rupiahs, 42 Malaysian ringgits, and 36 Singapore dollars. Suppose the foreign exchange rates are U.S. \(\$ 0.03\) for one baht, U.S. \(\$ 0.00011\) for one rupiah, U.S. \(\$ 0.294\) for one Malaysian ringgit, and U.S. \(\$ 0.656\) for one Singapore dollar. a. Write a row matrix \(A\) giving the value of the various currencies that Ethan holds. (Note: The answer is \(n o t\) unique.) b. Write a column matrix \(B\) giving the exchange rates for the various currencies. c. If Ethan exchanges all of his foreign currencies for U.S. dollars, how many dollars will he have?
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