Question

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

29. \(A\) is a \(3 \times 3\) matrix with three pivot positions.

Short answer

a. The equation \(Ax = 0\) has no free variable. Thus, the equation \(Ax = 0\)does not have a nontrivial solution.

b. The equation \(Ax = b\) has a solution for every possible \(b\).

Step-by-step solution

Determine the basic variable and free variable of the matrix

The variables corresponding to pivot columns in the matrix are calledbasic variables.The other variable is called a free variable.

\(3 \times 3\) matrix has one free variable and no basic variables.

Determine whether the equation \(Ax = b\) has a nontrivial solution

(a)

The homogeneous equation \(Ax = 0\) will have anontrivial solutionif and only if the equation has at least one free variable. The system will have anontrivial solution if a column in the coefficient matrix does not construct a pivot column.

If\(A\) is a \(3 \times 3\) matrix with three pivot positions, then the equation \(Ax = 0\) has no free variable. Therefore, the equation \(Ax = 0\) does not have a nontrivial solution.

Determine whether the equation \(Ax = b\) has at least one solution for every possible \({\mathop{\rm b}\nolimits} \)

(b)

Theorem 4tells us that for each \({\mathop{\rm b}\nolimits} \) in \({\mathbb{R}^m}\) , the equation \(Ax = b\) has a solution.

\(A\)has a pivot position in each of its three rows since it has three pivot positions.

According to theorem 4, the equation \(Ax = b\) has a solution for every value of \({\mathop{\rm b}\nolimits} \).

Thus, the equation \(Ax = b\) has a solution for every possible \(b\).