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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. (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

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

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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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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