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Chapter 2: Problem 2

The sizes of matrices \(A\) and \(B\) are given. Find the size of \(A B\) and \(B A\) whenever they are defined. \(A\) is of size \(3 \times 4\), and \(B\) is of size \(4 \times 3\).

Short Answer

Expert verified
The size of \(AB\) is \(3 \times 3\) and the size of \(BA\) is \(4 \times 4\).

Step by step solution

01

Matrix Multiplication Requirements

To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is met, then the product matrix exists, and its size will be the number of rows of the first matrix times the number of columns of the second matrix.
02

Check if \(AB\) is defined

To check if \(AB\) is defined, we need to compare the number of columns in \(A\) with the number of rows in \(B\). The matrix \(A\) is of size \(3 \times 4\), so it has 3 rows and 4 columns. The matrix \(B\) is of size \(4 \times 3\), so it has 4 rows and 3 columns. Since the number of columns in \(A\) (4) matches the number of rows in \(B\) (4), \(AB\) is defined.
03

Find the size of \(AB\)

Since \(AB\) is defined, we can now find its size. The size of the product matrix will be the number of rows in \(A\) times the number of columns in \(B\). \(A\) has 3 rows and \(B\) has 3 columns, so the size of \(AB\) will be \(3 \times 3\).
04

Check if \(BA\) is defined

To check if \(BA\) is defined, we need to compare the number of columns in \(B\) with the number of rows in \(A\). The matrix \(B\) is of size \(4 \times 3\), so it has 3 columns. The matrix \(A\) is of size \(3 \times 4\), so it has 3 rows. Since the number of columns in \(B\) (3) matches the number of rows in \(A\) (3), \(BA\) is defined.
05

Find the size of \(BA\)

Since \(BA\) is defined, we can now find its size. The size of the product matrix will be the number of rows in \(B\) times the number of columns in \(A\). \(B\) has 4 rows and \(A\) has 4 columns, so the size of \(BA\) will be \(4 \times 4\).
06

Summary

The sizes of matrices \(AB\) and \(BA\) are given as follows: - \(AB\) is of size \(3 \times 3\) - \(BA\) is of size \(4 \times 4\)

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Most popular questions from this chapter

Write the given system of linear equations in matrix form. $$ \begin{array}{r} -x_{1}+x_{2}+x_{3}=0 \\ 2 x_{1}-x_{2}-x_{3}=2 \\ -3 x_{1}+2 x_{2}+4 x_{3}=4 \end{array} $$

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?

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a \(2 \times 4\) matrix and \(B\) is a matrix such that \(A B A\) is defined, then the size of \(B\) must be \(4 \times 2\).

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{rr} 2 x+4 y-6 z= & 38 \\ x+2 y+3 z= & 7 \\ 3 x-4 y+4 z= & -19 \end{array} $$

Mr. Cross. Mr. Jones, and Mr. Smith each suffer from coronary heart disease. As part of their treatment, they were put on special low-cholesterol diets: Cross on diet I, Jones on diet II, and Smith on diet III. Progressive records of each patient's cholesterol level were kept. At the beginning of the first, second, third, and fourth months, the cholesterol levels of the three patients were: Cross: \(220,215,210\), and 205 Jones: \(220,210,200\), and 195 Smith: \(215,205,195\), and 190 Represent this information in a \(3 \times 4\) matrix.
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