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

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

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter12: Algebra Of Matrices
Section12.1: Algebra Of 2 X 2 Matrices
Problem 45PS
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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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