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Chapter 2: Problem 3

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 \(1 \times 7\), and \(B\) is of size \(7 \times 1\).

Short Answer

Expert verified
The size of the product AB is \(1 \times 1\), and the size of the product BA is \(7 \times 7\).

Step by step solution

01

Check if A and B can be multiplied

Matrix A has a size of \(1 \times 7\) and matrix B has a size of \(7 \times 1\). Since the number of columns in matrix A is equal to the number of rows in matrix B, matrix A and matrix B can be multiplied.
02

Determine the size of AB

The resulting matrix of product AB will have the number of rows of matrix A and the number of columns of matrix B. Since matrix A has 1 row and matrix B has 1 column, the product AB will have a size of \(1 \times 1\).
03

Check if B and A can be multiplied

Matrix B has a size of \(7 \times 1\) and matrix A has a size of \(1 \times 7\). Since the number of columns in matrix B is equal to the number of rows in matrix A, matrix B and matrix A can be multiplied.
04

Determine the size of BA

The resulting matrix of product BA will have the number of rows of matrix B and the number of columns of matrix A. Since matrix B has 7 rows and matrix A has 7 columns, the product BA will have a size of \(7 \times 7\).
05

Final Answer

The size of the product AB is \(1 \times 1\), and the size of the product BA is \(7 \times 7\).

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Most popular questions from this chapter

The property damage claim frequencies per 100 cars in Massachusetts in the years 2000,2001 , and 2002 are \(6.88,7.05\), and \(7.18\), respectively. The corresponding claim frequencies in the United States are 4.13, \(4.09\), and \(4.06\), respectively. Express this information using a \(2 \times 3\) matrix.

Bond Brothers, a real estate developer, builds houses in three states. The projected number of units of each model to be built in each state is given by the matrix $$ \begin{array}{l} \text { Model }\\\ \begin{array}{l} 111 \\ 120 \end{array}\\\ \begin{array}{r} \text { N.Y. } \\ A=\text { Conn. } \\ \text { Mass. } \end{array}\left[\begin{array}{rrrr} 60 & 80 & 120 & 40 \\ 20 & 30 & 60 & 10 \\ 10 & 15 & 30 & 5 \end{array}\right] \end{array} $$ The profits to be realized are \(\$ 20,000, \$ 22,000, \$ 25,000\), and \(\$ 30,000\), respectively, for each model I, II, III, and IV house sold. a. Write a column matrix \(B\) representing the profit for each type of house. b. Find the total profit Bond Brothers expects to earn in each state if all the houses are sold.

A dietitian plans a meal around three foods. The number of units of vitamin A, vitamin \(\mathrm{C}\), and calcium in each ounce of these foods is represented by the matrix \(M\), where $$ \begin{array}{l} \text { Food I } & \text { Food II } & \text { Food III } \\ \text { Vitamin A } & {\left[\begin{array}{rrr} 400 & 1200 & 800 \\ M= & \text { Vitamin C } \\ \text { Calcium } \end{array}\right.} & \begin{array}{rr} 110 \\ 90 \end{array} & \begin{array}{r} 570 \\ 30 \end{array} & \left.\begin{array}{r} 340 \\ 60 \end{array}\right] \end{array} $$ The matrices \(A\) and \(B\) represent the amount of each food (in ounces) consumed by a girl at two different meals, where \(\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}\) $$ A=\left[\begin{array}{lll} 7 & 1 & 6 \end{array}\right] $$ \(\begin{array}{lll}\text { Food I } & \text { Food II } & \text { Food III }\end{array}\) $$ B=\left[9 \quad \left[\begin{array}{ll} 9 & 3 \end{array}\right.\right. $$ $$ 2] $$ Calculate the following matrices and explain the meaning of the entries in each matrix. a. \(M A^{T}\) b. \(M B^{T}\) c. \(M(A+B)^{T}\)

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. $$ \begin{array}{l} \begin{array}{r} x_{1}+x_{2}+x_{3}+x_{4}=b_{1} \\ x_{1}-x_{2}-x_{3}+x_{4}=b_{2} \\ x_{2}+2 x_{3}+2 x_{4}=b_{3} \\ x_{1}+2 x_{2}+x_{3}-2 x_{4}=b_{4} \end{array}\\\ \text { where (i) } b_{1}=1, b_{2}=-1, b_{3}=4, b_{4}=0\\\ \text { and } \quad \text { (ii) } b_{1}=2, b_{2}=8, b_{3}=4, b_{4}=-1 \end{array} $$

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