In Gaussian elimination with partial pivoting, show that if we cannot find a nonzero pivot then the matrix is singular.  Let L be lower triangular and let eT k = XXXXXXXXXXa) Write an algorithm to...

In Gaussian elimination with partial pivoting, show that if we cannot find a nonzero pivot then the matrix is singular.

 Let L be lower triangular and let eT k = (0 ...1... 0). (a) Write an algorithm to solve Lx = ek. (b) Where are the known zeros in the solution vector? (c) What columns of L are not needed? (d) Write an algorithm to overwrite L with L−1 .