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(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.

(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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do Exercise 1 part C and D and please show step by step and explain

Topic: Magic Squares

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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