. For each part, nd a specic example of a matrix satisfying the given properties. You must show that your example satises the described properties. [a] A matrix A1 such that there exists a permutation matrix P, a lower triangular matrix L, and an upper triangular matrix U such that PA1 = LU , but there does not exist a lower triangular L1] and upper triangular U0 such that A1 = LoUg. (h) A matrix Ag such that there exists a lower triangular L and upper triangular U with A2 = LU , but there does ngt exist a permutation matrix P 79 1', lower triangular matrix Lo, and upper triangular matrix Ug such that PA; = LgUg. (c) A 3 x 3 matrix A3 such that for every permutation matrix P (Note. There are 6 of them], there exists a lower triangular matrix L and an upper triangular matrix U such that PA3 = LU. (d) A singular matrix A; such that there exists a lower triangular matrix L and a unit upper triangular matrix U such that A; = LU. (e) A matrix A5 such that there is no way to reduce A5 to an upper triangular matrix using only row operations of the form EiJ-(c) with i :9- j, but there exists a lower triangular matrix L and an upper triangular matrix U such that A5 = LU