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Exercise Prove that for any square matrix A, there are unique symmetric matrix X and skew-symmetric matrix Y such that A = X + Y.

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Exercise Prove that if A is invertible, then AT is also invertible, and (AT)-1 = (A-1)T.

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Exercise By making use of the computation in this exercise, solve the following systems.

2x1 + 7x2 = 2
3x1 + 9x2 = -1
x1 + 2x2 + 4x3 = 1
  - x2 - x3 = 0
2x1 + 3x2 + 8x3 = -1
2x1 + x2   + x4 = 1
  x2 + x3 - x4 = 0
2x1 - x2 - 2x3 + 2x4 = 1
x1   - x3 + x4 = 0

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