Exercise Prove that for any square matrix A, there are unique symmetric matrix X and skew-symmetric matrix Y such that A = X + Y.
Exercise Prove that if A is invertible, then AT is also invertible, and (AT)-1 = (A-1)T.
Exercise By making use of the computation in this exercise, solve the following systems.
|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|