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1 Let M be a 3×3 magic square with magic number s. (a) Prove that the sum of M's entries is 3s. (b) Prove that s = 3. m2,2. (c) Prove that m2,2 is the average of the entries in its row, its column, and in each diagonal.

1 Let M be a 3×3 magic square with magic number s. (a) Prove that the sum of M's entries is 3s. (b) Prove that s = 3. m2,2. (c) Prove that m2,2 is the average of the entries in its row, its column, and in each diagonal.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Please do Exercise 1 part A,B,C,D and please show step by step and explain

1 Let M be a 3×3 magic square with magic number s. (a) Prove that the sum of M's entries is 3s. (b) Prove that s = 3. m2, 2. (c) Prove that m2,2 is the average of the entries in its row, its column, and in each diagonal. (d) Prove that m2,2 is the median of M's entries. 2 Solve the system a+b = s, c+d=s, a+c=s, b+d=s, a+d=s, and b+c=s. 3 Show that dim M2,0 = 0. 4 Let the trace function be Tr(M) = m₁,1 + · +mn,n. Define also the sum down the other diagonal Tr* (M) = m1,n + ··· + Mn, 1. (a) Show that the two functions Tr, Tr*: Mnxn → R are linear. (b) Show that the function 0: Mnxn → R² given by 0(M) = (Tr(M), Tr* (m)) is linear. (c) Generalize the prior item.
Transcribed Image Text:1 Let M be a 3×3 magic square with magic number s. (a) Prove that the sum of M's entries is 3s. (b) Prove that s = 3. m2, 2. (c) Prove that m2,2 is the average of the entries in its row, its column, and in each diagonal. (d) Prove that m2,2 is the median of M's entries. 2 Solve the system a+b = s, c+d=s, a+c=s, b+d=s, a+d=s, and b+c=s. 3 Show that dim M2,0 = 0. 4 Let the trace function be Tr(M) = m₁,1 + · +mn,n. Define also the sum down the other diagonal Tr* (M) = m1,n + ··· + Mn, 1. (a) Show that the two functions Tr, Tr*: Mnxn → R are linear. (b) Show that the function 0: Mnxn → R² given by 0(M) = (Tr(M), Tr* (m)) is linear. (c) Generalize the prior item.
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