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).

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