91Ó°ÊÓ

Use this information for Exercises 1-8. Troy Aikman, Randall Cunningham, and Steve Young were top-performing quarterbacks in the National Football League throughout their careers. The rows in matrix \([A]\) and matrix \([B]\) show data for Aikman, Cunningham, and Young, in that order. The columns show the number of passing attempts, pass completions, touchdown passes, and interceptions, from left to right. Matrix \([A]\) shows stats from 1992 , and matrix \([B]\) shows stats from \(1998 .\) $$ [A]=\left[\begin{array}{rrrr} 473 & 302 & 23 & 14 \\ 384 & 233 & 19 & 11 \\ 402 & 268 & 25 & 7 \end{array}\right] \quad[B]=\left[\begin{array}{llll} 315 & 187 & 12 & 5 \\ 425 & 259 & 34 & 10 \\ 517 & 322 & 36 & 12 \end{array}\right] $$ Is \([A]+[B]\) equal to \([B]+[A] ?\) Do you think this result is always true for matrix addition? Explain.

Step by step solution

01

Understanding Matrix Addition

To find the sum of the two matrices \([A]\) and \([B]\), we perform matrix addition. In matrix addition, each element of one matrix is added to the corresponding element of the other matrix.

02

Add Matrices [A] and [B]

Let's add the corresponding elements of matrices \([A]\) and \([B]\): \[[A]+[B] = \begin{bmatrix}473 + 315 & 302 + 187 & 23 + 12 & 14 + 5 \384 + 425 & 233 + 259 & 19 + 34 & 11 + 10 \402 + 517 & 268 + 322 & 25 + 36 & 7 + 12\end{bmatrix}\]This results in:\[= \begin{bmatrix}788 & 489 & 35 & 19 \809 & 492 & 53 & 21 \919 & 590 & 61 & 19\end{bmatrix}\]

03

Add Matrices [B] and [A]

Now, we add the corresponding elements of matrices \([B]\) and \([A]\):\[[B]+[A] = \begin{bmatrix}315 + 473 & 187 + 302 & 12 + 23 & 5 + 14 \425 + 384 & 259 + 233 & 34 + 19 & 10 + 11 \517 + 402 & 322 + 268 & 36 + 25 & 12 + 7\end{bmatrix}\]This also results in:\[= \begin{bmatrix}788 & 489 & 35 & 19 \809 & 492 & 53 & 21 \919 & 590 & 61 & 19\end{bmatrix}\]

04

Compare [A] + [B] and [B] + [A]

Both \([A]+[B]\) and \([B]+[A]\) result in the same matrix:\[= \begin{bmatrix}788 & 489 & 35 & 19 \809 & 492 & 53 & 21 \919 & 590 & 61 & 19\end{bmatrix}\]The sums are equal, confirming the commutative property of matrix addition.

05

Deduce General Property

Matrix addition is commutative, meaning \([A] + [B] = [B] + [A]\) for any conformable matrices \([A]\) and \([B]\). This is always true regardless of the specific numbers involved, as long as the matrices are of the same size.

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Key Concepts

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

Commutative Property

The commutative property is a fundamental principle in mathematics, applicable to different operations, including matrix addition. This property states that the order in which numbers (or matrices) are added does not affect the resulting sum. In simpler terms, if you switch the order of the matrices in addition, the result remains the same.

For two matrices

• Let and be two conformable matrices (meaning they have the same dimensions), the commutative property states that: \[ [A] + [B] = [B] + [A] \]


• In the given exercise, this property was demonstrated by adding matrix from 1992 stats and matrix from 1998 stats.

Each pair of matrices, when added in different orders, resulted in the same resulting matrix. Thus, ensuring this property holds for matrix addition.
Understanding this property builds a solid foundation for working with more complex matrix operations and ensures mathematical proofs involving matrices are consistent and logical.

Matrix Operations

Matrix operations include a range of mathematical procedures that can be performed on matrices, such as addition, subtraction, scaling, and multiplication.

In matrix addition:

• Each element in one matrix is added to the corresponding element in another matrix. This requires both matrices to have the same dimensions (number of rows and columns).
• In the exercise, the matrices and both have the same dimensions of 3x3, allowing them to be added together without issue.

For the addition of matrices and , each element's operation was simple, involving pairwise addition:
Example:

• For the (1,1) position found in the resulting matrix, it is the sum of 473 (from matrix ) and 315 (from matrix ), resulting in 788.

This method applies to all elements of the matrices. This basic understanding of matrix operations is crucial for solving more complex equations and understanding operations beyond simple addition, such as matrix multiplication or finding inverses.

Mathematical Proofs

Mathematical proofs play a crucial role in confirming the general properties of mathematical concepts, such as the commutative property in matrix addition. A mathematical proof offers a logical argument that verifies a particular theorem or statement is universally true.

When exploring whether <[A> + = + ?, we used a mathematical process:

• Compute both + and + , demonstrating through calculation that both result in the same matrix.
• This ensures beyond individual examples, that the property holds for all instances meeting the criteria (same-sized matrices).

These proofs often rely on logical steps and repetition to confirm understanding. Ensuring mathematical proofs are thorough and clear helps avoid errors in more extensive computations and theoretical applications. It is the backbone of verifying concepts that are then applied in various fields like computer science, engineering, and economics.