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Chapter 8: Problem 17

Perform the given operations (if defined) on the matrices. $$A=\left[\begin{array}{rrr}1 & -3 & \frac{1}{3} \\\5 & 0 & -2\end{array}\right], \quad B=\left[\begin{array}{rr}8 & 0 \\\3 & -2 \\\2 & -6\end{array}\right], \quad C=\left[\begin{array}{rr}-4 & 5 \\\0 & 1 \\\\-2 & 7 \end{array}\right]$$If an operation is not defined, state the reason. $$A+2 B$$

Short Answer

Expert verified
The operation \(A + 2B\) is not defined because the matrices A and B do not have the same dimensions. Matrix addition or subtraction can only be performed on matrices of same dimensions.

Step by step solution

01

Understand the Matrix dimensions

Matrix A is a 2x3 matrix (2 rows and 3 columns), Matrix B is a 3x2 matrix (3 rows and 2 columns). Matrix addition/subtraction requires that both matrices have the same dimensions. The dimensions of matrix A and matrix B are not the same. Therefore the operation \(A+2B\) is not defined.
02

Confirming Non-Coformability for Addition

Addition of matrices is sessionally feasible only when both matrices have the same number of rows and columns. Since the number of rows and columns in Matrix A does not match the number of rows and columns in Matrix B, we conclude that the operation \(A+2B\) is not possible and cannot be defined in terms of matrix operations.

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Key Concepts

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

Matrix Addition
Matrix addition is a fundamental operation where two matrices are added together. This process involves adding corresponding elements from each matrix. If you think of each element in the matrix as a "cell" in a grid, you'll be adding the numbers in each matching "cell." The result is a new matrix, where each element is the sum of the elements from the two matrices being added.
A crucial rule in matrix addition is that the matrices must have identical dimensions, meaning they need the same number of rows and columns. Without matching dimensions, the addition is not possible. Consider two matrices: - Matrix A: A 2x2 - Matrix B: Also a 2x2 These matrices can be added because they share the same size. Each element from the first matrix aligns perfectly with an element from the second matrix, allowing you to add each pair of corresponding elements.
If the dimensions do not match, like in our given problem with matrices A and B, addition is undefined because there's no one-to-one correspondence between the elements of the two matrices.
Matrix Dimensions
Matrix dimensions describe the size of a matrix, given as "rows by columns." Understanding matrix dimensions is vital in operations like addition, where compatibility is strictly enforced.
Let's revisit our matrices: - Matrix A is a 2x3, having two rows and three columns - Matrix B is a 3x2, with three rows and two columns By identifying these dimensions, we quickly realize that A has a different shape compared to B. This difference in shape is what makes certain operations, like their addition, impossible.
Matrix dimensions are not just numbers; they provide a picture of how a matrix looks and even suggest allowable operations. Always check and double-check dimensions before attempting matrix operations.
This understanding helps not only in performing valid operations but also in spotting errors in problem-solving.
Conformable Matrices
Conformable matrices refer to matrices that meet specific conditions for certain operations. Specifically, two matrices are said to be conformable for addition if they have the same size, i.e., the same number of rows and columns.
Why is this important? When matrices are conformable, operations can be performed without concern for missing or extra elements. Consider conformable matrices as perfect partners; their matching dimensions ensure that every element has a clear counterpart. - Matrix A: 2 rows and 2 columns - Matrix B: 2 rows and 2 columns These are conformable for addition, ensuring each cell in a matrix aligns with a counterpart.
In scenarios where matrices aren't conformable, like our example matrices A (2x3) and B (3x2), operations like addition can't proceed. Understanding conformability is essential for correctly applying matrix operations, avoiding mistakes, and solving mathematical problems effectively.
Checking for conformability acts as a filter, allowing only valid operations that fit well with the given matrices.

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Most popular questions from this chapter

An airline charges 380 dollar for a round-trip flight from New York to Los Angeles if the ticket is purchased at least 7 days in advance of travel. Otherwise, the price is 700 dollar . If a total of 80 tickets are purchased at a total cost of 39,040 dollar, find the number of tickets sold at each price.

In this set of exercises, you will use the method of solving linear systems using matrices to study real-world problems. A grocery store carries two brands of diapers. For a certain week, the number of boxes of Brand \(A\) diapers sold was 4 more than the number of boxes of Brand B diapers sold. Brand A diapers cost \(\$ 10\) per box and Brand B diapers cost \(\$ 12\) per box. If the total revenue generated that week from the sale of diapers was \(\$ 172,\) how many of each brand did the store sell?

Involve positive-integer powers of a square matrix \(A . A^{2}\) is defined as the product \(A A ;\) for \(n \geq 3, A^{n}\) is defined as the product \(\left(A^{n-1}\right) A\) Find \(\left(A^{2}\right)^{-1}\) and \(\left(A^{-1}\right)^{2},\) where \(A=\left[\begin{array}{rr}1 & -2 \\ -1 & 3\end{array}\right] .\) What do you observe?

This set of exercises will draw on the ideas presented in this section and your general math background. Find the inverse of $$\left[\begin{array}{lll}a & a & a \\\0 & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$ where \(a\) is nonzero. Evaluate this inverse for the case in which \(a=1\)

An electronics firm makes a clock radio in two different models: one (model 380 ) with a battery backup feature and the other (model 360 ) without. It takes 1 hour and 15 minutes to manufacture each unit of the model 380 radio, and only 1 hour to manufacture each unit of the model \(360 .\) At least 500 units of the model 360 radio are to be produced. The manufacturer realizes a profit per radio of \(\$ 15\) for the model 380 and only \(\$ 10\) for the model \(360 .\) If at most 2000 hours are to be allocated to the manufacture of the two models combined, how many of each model should be made to maximize the total profit?
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