According to the Wikipedia entry link below, the simplex tableau is the matrix of the constraint equations, along with artificial variables. Here is how you tell if the tableau is not in final form - if it is not in row-echelon form (from linear algebra). For this particular type of matrix, you need to have at least one column with all zeros except for a row, for each variable. For instance, this would be a matrix in which there is a column with all zeros except one, for each of the variables. A matrix with all zeros except diagonally (where there are ones) is an identity matrix. You have to keep using linear operations (multiply a row by a constant, combine two rows, etc.) until enough of the columns are all zeros except with a 1 entry at the row denoting the variable. You select pivot columns that work best for you. It is best to select the pivot column that will reduce other row entries to zero, so that gets you closer to having an identity matrix. Do you also understand why there are artificial variables? It has to do with rewriting the equation from inequalities to a longer equation with an equal sign and a few more variables.