Amatrix is skew symmetric of a^t = -a. prove that if b is a square matrix that 1/2(b - b^t) is skew symmetric and 1/2(b + b^t) is symmetric. find the inverse of a = [1 2 0 2 1 0 0 0 3] if it exists. a) give a 2 times 2 example that proves that the sum of two invertible matrices need not be invertible. b) give a 2 times 2 example that proves that the sum of two singular matrices need not be singular. let a be an invertible matrix. prove that if ab is defined then rank ab = rank b and if ba is defined that rank ba = rank b.

the sum of the lengths of any two sides of a triangle is greater than the length of the third side. if you take the three sides of a triangle and add them in pairs, the sum is greater than (not equal to) the third side. if that is not true, then it is not possible to construct a triangle with the given side lengths.

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answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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