Proof That Matrix Addition is Associative

TL;DR

Matrix addition is associative, meaning that (A + B) + C = A + (B + C).

Transcript

in this problem we're going to prove that matrix addition is associative so we're going to prove that a plus b plus c is the same as a plus b plus c and this is for any matrices abc of the same size so proof so we need some notation in order to indicate the ij entry of each matrix so let a sub i j so a i j denote the i j entry of a b i j or b sub i... Read More

Key Insights

• ❓ The proof demonstrates that matrix addition follows the associative property.
• 😃 Notation is used to indicate the i-j entry of each matrix, facilitating the analysis.
• 🙃 Both sides of the equation are examined by considering the sum of individual entries.
• 🏑 The associativity of addition in a field is utilized to conclude the equality of the matrix entries.

Questions & Answers

Q: What is the goal of the proof?

The goal is to prove that matrix addition is associative for matrices A, B, and C of the same size.

Q: How is the left-hand side (LHS) of the equation analyzed?

The LHS is analyzed by considering the individual entries of matrices A, B, and C and their respective sums.

Q: How is the right-hand side (RHS) of the equation examined?

The RHS is examined by looking at the entries of matrices B and C separately and their sum.

Q: How does the proof establish associativity?

By showing that the entries of both sides of the equation are equal, it is concluded that matrix addition is associative.

Summary & Key Takeaways

• The goal is to prove that matrix addition is associative for all matrices A, B, and C of the same size.

• The left-hand side (LHS) is analyzed by considering the individual entries of A, B, and C.

• The right-hand side (RHS) is also examined by looking at the entries of B and C separately.

• By showing that the entries of both sides are equal, it is concluded that matrix addition is associative.