Let m be an n × n matrix with each entry equal to either 0 or 1. let mij denote the entry in row i and column j. a diagonal entry is one of the form mii for some i. swapping rows i and j of the matrix m denotes the following action: we swap the values mik and mjk for k = 1, 2, . . , n. swapping two columns is defined analogously. we say that m is rearrangeable if it is possible to swap some of the pairs of rows and some of the pairs of columns (in any sequence) so that, after all the swapping, all the diagonal entries of m are equal to 1. (a) give an example of a matrix m that is not rearrangeable, but for which at least one entry in each row and each column is equal to 1. (b) give a polynomial-time algorithm that determines whether a matrix m with 0-1 entries is rearrangeable.

Two spray paint machines are used to paint different portions of a large wall. the first machine (nicknamed "paint pro") is used for two hours. the second machine (nicknamed "goldilocks") is used for an hour and a half. when they are working at the same time, they can paint 55 square feet per minute. together they painted 5850 square feet of wall. how many square feet of wall per minute can each machine paint? e used to paint different portions of a large wall. the first machine (nicknamed "paint pro") is used for two hours. the second machine (nicknamed "goldilocks") is used for an hour and a half. when they are working at the same time, they can paint 55 square feet per minute. together they painted 5850 square feet of wall. how many square feet of wall per minute can each machine paint?