In Exercises 29–32, (a) does the equation have a nontrivial solution and (b) does the equation \(A{\bf{x}} = {\bf{b}}\) have at least one solution for every possible b?

31. A is a \(3 \times 2\) matrix with two pivot positions

It is given that the order of matrix Ais . This means that there are 3 rows \(A{\bf{x}} = {\bf{b}}\)and 2 columns. Also, it is given that there are two pivot positions.

So, not all rows have a pivot element (each column serves as a pivot column), and one row should be empty.

Matrix Aof \(3 \times 2\) order in the augmented form \(\left[ {\begin{array}{*{20}{c}}A&0\end{array}} \right]\) is written as shown below:

It is observed that there are no free variables in \(A{\bf{x}} = 0\).

Thus, the system of equations does not have a nontrivial solution.

Here, not all rows have a pivot element (the last row does not have a pivot position).

Consider vector \(A{\bf{x}} = 0\)\({\bf{b}} = \left[ {\begin{array}{*{20}{c}}{{b_1}}\\{{b_2}}\\{{b_3}}\end{array}} \right]\).

Matrix Aof \(3 \times 2\) order in the augmented form \(\left[ {\begin{array}{*{20}{c}}A&{\bf{b}}\end{array}} \right]\) is written as shown below:

It is observed that there are possible values of b in . Also, there are three rows with two pivot positions. So, it cannot have a solution for every possible b.

Thus, the system of equations does not have a solution for every possible b.