7. Establish for A is m × n and B, C are n x p, that A(B+C) = AB + AC

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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Chapter3: Matrices
Section3.1: Matrix Operations
Problem 20EQ: Referring to Exercise 19, suppose that the unit cost of distributing the products to stores is the...
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### Problem Statement

7. Establish for \( A \) is \( m \times n \) and \( B, C \) are \( n \times p \), that \( A(B + C) = AB + AC \).

---

This problem requires you to demonstrate the distributive property of matrix multiplication. You must show that multiplying a matrix \( A \) by the sum of two matrices \( B \) and \( C \) results in the same matrix as multiplying \( A \) by \( B \) and \( A \) by \( C \) separately, and then adding the results.

**Explanation:**

- **Matrix Dimensions:**
  - \( A \) is an \( m \times n \) matrix.
  - \( B \) and \( C \) are both \( n \times p \) matrices.

- **Operation:**
  - \( A(B + C) \) means that you first perform the matrix addition \( B + C \), which is defined because \( B \) and \( C \) have the same dimensions, and then multiply the resulting matrix by \( A \).
  - The result will be an \( m \times p \) matrix.

- **Distributive Property:**
  - You need to show that this operation is the same as calculating \( AB \) and \( AC \) (which are also \( m \times p \) matrices) and then adding them together.

This property is a fundamental aspect of matrix algebra and is used extensively in various applications involving linear transformations and systems of equations.
Transcribed Image Text:### Problem Statement 7. Establish for \( A \) is \( m \times n \) and \( B, C \) are \( n \times p \), that \( A(B + C) = AB + AC \). --- This problem requires you to demonstrate the distributive property of matrix multiplication. You must show that multiplying a matrix \( A \) by the sum of two matrices \( B \) and \( C \) results in the same matrix as multiplying \( A \) by \( B \) and \( A \) by \( C \) separately, and then adding the results. **Explanation:** - **Matrix Dimensions:** - \( A \) is an \( m \times n \) matrix. - \( B \) and \( C \) are both \( n \times p \) matrices. - **Operation:** - \( A(B + C) \) means that you first perform the matrix addition \( B + C \), which is defined because \( B \) and \( C \) have the same dimensions, and then multiply the resulting matrix by \( A \). - The result will be an \( m \times p \) matrix. - **Distributive Property:** - You need to show that this operation is the same as calculating \( AB \) and \( AC \) (which are also \( m \times p \) matrices) and then adding them together. This property is a fundamental aspect of matrix algebra and is used extensively in various applications involving linear transformations and systems of equations.
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