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Problem 10 Let 8 : M2×2(F) → F be a function with the following three properties. (1) 8 is a linear function of each row of the matrix when the other row is held fixed. (2) If the two rows of A € M2×2(F) are identical, then 8(A) = 0. (3) If I is the 2 × 2 identity matrix, then §(I) = 1. Prove that √(A) = det(A) for all A € M2×2(F).

Problem 10 Let 8 : M2×2(F) → F be a function with the following three properties. (1) 8 is a linear function of each row of the matrix when the other row is held fixed. (2) If the two rows of A € M2×2(F) are identical, then 8(A) = 0. (3) If I is the 2 × 2 identity matrix, then §(I) = 1. Prove that √(A) = det(A) for all A € M2×2(F).

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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Problem 10 Let 8: M2x2(F) → F be a function with the following three properties. (1) 8 is a linear function of each row of the matrix when the other row is held fixed. (2) If the two rows of A € M2x2(F) are identical, then (A) = 0. (3) If I is the 2 × 2 identity matrix, then 8(I) = 1. Prove that 8(A) = det(A) for all A ¤ M2×2(F).
Transcribed Image Text:Problem 10 Let 8: M2x2(F) → F be a function with the following three properties. (1) 8 is a linear function of each row of the matrix when the other row is held fixed. (2) If the two rows of A € M2x2(F) are identical, then (A) = 0. (3) If I is the 2 × 2 identity matrix, then 8(I) = 1. Prove that 8(A) = det(A) for all A ¤ M2×2(F).
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