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Chapter 9: Problem 80

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with two matrices that can be multiplied but not added.

Short Answer

Expert verified
The statement makes sense considering the fundamental rules of matrix addition and multiplication. If the dimensions (number of rows and columns) of two matrices are different but the second dimension of the first matrix equals the first dimension of the second, they can be multiplied but not added.

Step by step solution

01

Understanding Matrix Addition

Two matrices can be added or subtracted only if they are of the same size, i.e., they must have the same number of rows and the same number of columns.
02

Understanding Matrix Multiplication

Two matrices can be multiplied if and only if the number of columns in the first matrix equals the number of rows in the second matrix.
03

Apply the Above Understandings to the Original Statement

When the statement 'I'm working with two matrices that can be multiplied but not added' is evaluated using the aforementioned rules, it is found to make sense. This situation can happen if the two matrices have different dimensions but satisfy the condition for multiplication. For instance, a 3x2 matrix can be multiplied with a 2x3 matrix but they cannot be added as they don't have the same dimensions.

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Most popular questions from this chapter

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ A(B+C) $$

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. A furniture company produces three types of desks: a children's model, an office model, and a deluxe model. Each desk is manufactured in three stages: cutting, construction, and finishing. The time requirements for each model and manufacturing stage are given in the following table. $$ \begin{array}{ccc} {} & {\text { Children's }} & {\text { Office }} & {\text { Deluxe }} \\\& {\text { Model }} & {\text { Model }} & {\text { Model }} \\ {\text { Cutting }} & {2 \text { hr }} & {3 \text { hr }} & {2 \text { hr }} \\\ {\text { Construction }} & {2 \text { hr }} & {1 \text { hr }} & {3 \text { hr }} \\ {\text { Finishing }} & {1 \text { hr }} & {1 \text { hr }} & {2 \text { hr }} \end{array} $$ Each week the company has available a maximum of 100 hours for cutting, 100 hours for construction, and 65 hours for finishing. If all available time must be used, how many of each type of desk should be produced each week?

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rr} {-2} & {3} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{ll} {8} & {1} \\ {5} & {4} \end{array}\right] $$

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} w+x+y+z &=4 \\ 2 w+x-2 y-z &=0 \\ w-2 x-y-2 z &=-2 \\ 3 w+2 x+y+3 z &=4 \end{aligned}\right. $$

Describe matrices that cannot be added or subtracted.
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