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Let A= [ array ccc 0 & 2 & -3 -2 & 0 & 1 3 & -1 & 0 array ] and B be a matrix such that B(I-A)=I+A . Then the sum of the diagonal elements of B ^ T B is equal to _ _ _ _ .

Correct answer

0

Step-by-step solution

A is skew-symmetric ( A^T = -A ). From B(I-A) = I+A : B = (I+A)(I-A)⁻¹ . B^T = ((I-A)^T)⁻¹(I+A)^T = (I+A)⁻¹(I-A) . Since A is skew-symmetric, (I-A) and (I+A) commute: (I-A)(I+A) = I - A^2 = (I+A)(I-A) . B^TB = (I+A)⁻¹(I-A)(I+A)(I-A)⁻¹ = (I+A)⁻¹(I+A)(I-A)(I-A)⁻¹ = I . Trace of B^TB = tr (I) = 3 .

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