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Chapter 2: Problem 22

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

Short Answer

Expert verified
The matrix product is: \( \left[\begin{array}{rrr} 3 & 27 & 6 \\ 21 & 15 & 0 \\ 6 & 6 & 6 \end{array}\right] \)

Step by step solution

01

Multiply the matrices by their scalar values

First, we need to multiply the given matrix by the scalar value 3. $$ 3\left[\begin{array}{rrr} 2 & -1 & 0 \\ 2 & 1 & 2 \\ 1 & 0 & -1 \end{array}\right] = \left[\begin{array}{rrr} 6 & -3 & 0 \\ 6 & 3 & 6 \\ 3 & 0 & -3 \end{array}\right] $$
02

Perform matrix multiplication

Now, we'll multiply the resulting matrix with the second matrix. $$ \left[\begin{array}{rrr} 6 & -3 & 0 \\ 6 & 3 & 6 \\ 3 & 0 & -3 \end{array}\right] \left[\begin{array}{rrr} 2 & 3 & 1 \\ 3 & -3 & 0 \\ 0 & 1 & -1 \end{array}\right] $$ To perform the matrix multiplication, recall that the value of the element in the i-th row and j-th column of the product is calculated by taking the dot product of the i-th row from the first matrix and the j-th column of the second matrix.
03

Compute each element of the product matrix

We will now compute the elements of the product matrix: - Element (1,1) = (6)(2) + (-3)(3) + (0)(0) = 12 - 9 + 0 = 3 - Element (1,2) = (6)(3) + (-3)(-3) + (0)(1) = 18 + 9 + 0 = 27 - Element (1,3) = (6)(1) + (-3)(0) + (0)(-1) = 6 + 0 + 0 = 6 - Element (2,1) = (6)(2) + (3)(3) + (6)(0) = 12 + 9 + 0 = 21 - Element (2,2) = (6)(3) + (3)(-3) + (6)(1) = 18 - 9 + 6 = 15 - Element (2,3) = (6)(1) + (3)(0) + (6)(-1) = 6 + 0 - 6 = 0 - Element (3,1) = (3)(2) + (0)(3) + (-3)(0) = 6 + 0 + 0 = 6 - Element (3,2) = (3)(3) + (0)(-3) + (-3)(1) = 9 + 0 - 3 = 6 - Element (3,3) = (3)(1) + (0)(0) + (-3)(-1) = 3 + 0 + 3 = 6 Now, we will put all these elements in the resulting 3x3 matrix: $$ \left[\begin{array}{rrr} 3 & 27 & 6 \\ 21 & 15 & 0 \\ 6 & 6 & 6 \end{array}\right] $$ The product of the given scalar and matrices is: $$ 3\left[\begin{array}{rrr} 2 & -1 & 0 \\ 2 & 1 & 2 \\ 1 & 0 & -1 \end{array}\right] \left[\begin{array}{rrr} 2 & 3 & 1 \\ 3 & -3 & 0 \\ 0 & 1 & -1 \end{array}\right] = \left[\begin{array}{rrr} 3 & 27 & 6 \\ 21 & 15 & 0 \\ 6 & 6 & 6 \end{array}\right] $$

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Key Concepts

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

Scalar Multiplication
Scalar multiplication is a fundamental operation in matrix algebra where each entry of a matrix is multiplied by a constant, known as a scalar. In the context of our exercise, the scalar is 3 and the matrix in question is a 3x3 matrix. Scalar multiplication affects each element independently, scaling the entire matrix without changing its dimensions. The result is a new matrix where each element has been multiplied by the scalar. For example, when we multiply the scalar 3 with the matrix, each entry in the matrix is tripled, leading to a new matrix which is used for further operations.
Dot Product
The dot product, also known as the scalar product, is a crucial concept in computing the elements of the product of two matrices. When multiplying two matrices, we calculate the dot product between the rows of the first matrix and the columns of the second matrix. The dot product is the sum of the products of corresponding entries from these rows and columns. For instance, to find the element in the first row and first column of the resulting matrix, we multiply the corresponding entries from the first row of the first matrix and the first column of the second matrix, then sum them up. This process is repeated for each element of the resulting matrix product. It's essential to ensure that the number of columns in the first matrix matches the number of rows in the second to perform the dot product.
Matrix Algebra
Matrix algebra encompasses various operations that can be performed on matrices, including addition, subtraction, multiplication, and scalar multiplication. It is an extension of arithmetic to matrices where these operations must follow specific rules that are sometimes counterintuitive. For instance, matrix multiplication is not commutative; the order in which you multiply matrices matters greatly. The product of two matrices is another matrix where each entry is computed using the dot product, as mentioned earlier. Matrix algebra is fundamental in many areas of mathematics and applied sciences, often used to solve systems of linear equations and represent linear transformations.
Elementary Operations
Elementary operations are basic manipulations that can be performed on the rows or columns of a matrix, which are used to solve systems of linear equations, among other applications. These operations include swapping rows or columns, multiplying a row or column by a non-zero scalar, and adding a scalar multiple of one row or column to another. These operations are powerful tools that help simplify matrices and are the foundation of more complex matrix manipulation techniques, such as finding the inverse or determinant of a matrix. In our exercise, scalar multiplication can be considered an elementary operation, as it involves multiplying each element of the matrix by the same scalar.

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Most popular questions from this chapter

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. The management of Hartman Rent-A-Car has allocated $$\$ 1.5$$ million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full- size cars. Compacts cost $$\$ 12,000$$ each, intermediatesize cars cost $$\$ 18,000$$ each, and full-size cars cost $$\$ 24,000$$ each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100 , determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

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?

Mortality actuarial tables in the United States were revised in 2001, the fourth time since 1858 . Based on the new life insurance mortality rates, \(1 \%\) of 60 -yr-old men, \(2.6 \%\) of 70 -yr-old men, \(7 \%\) of 80 -yr-old men, \(18.8 \%\) of 90 -yr-old men, and \(36.3 \%\) of 100 -yr-old men would die within a year. The corresponding rates for women are \(0.8 \%, 1.8 \%, 4.4 \%, 12.2 \%\), and \(27.6 \%\), respectively. Express this information using a \(2 \times 5\) matrix.

Find the matrix \(A\) if $$ A\left[\begin{array}{rr} 1 & 2 \\ 3 & -1 \end{array}\right]=\left[\begin{array}{rr} 2 & 1 \\ 3 & -2 \end{array}\right] $$

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{r} x-2 y=8 \\ 3 x+4 y=4 \end{array} $$
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