Solve the following problems showing any necessary work This

Solve the following problems, showing any necessary work. This includes row operations. How many solutions does each of the following systems of linear equations have? Circle the entries which led you to your conclusion If there are no solutions, circle the row that indicates that there are no solutions. If there is exactly one solution, circle the pivots. If there is more than one solution, circle the columns which do not have pivots in them.

Solution

We know that if a matrix is in row-echelon form, then the first nonzero entry of each row is called a pivot, and the columns in which pivots appear are called pivot columns.

(a) Here, all the columns are pivot columns. The matrix gives us 5 linear equations in 5 vriables. There will be a unique solution.

(b) The 3rd row indicates that 0 = 1 which is a contradiction. The system will not have any solution because of the 3rd row.

(c) The 2nd, 3rd and the 4th columns do not have any pivots.The 3rd row does not have any non-zero entry. The matrix gives us a system of 2 linear equations in 5 variables. Thus, there will be infinitely many solutions.