Question-3

Comment on the truth value of the following statements:

  1. Every system of linear equations has either a unique solution, no solution or infinitely many solutions.

  2. If each equation in a system of linear equations is multiplied by a non-zero constant \(c\) then the solution of the system is \(c\) times the solution of the old system.

  3. If \(Ax=b\) has a solution, then \(cAx=b\) where \(c\neq0\) will also have a solution.

  4. If \(Ax=b\) has a solution, then \(\frac{1}{c}Ax=b\) where \(c\neq0\) will also have a solution.

Statement-1

Geometrically, it is quite clear that these are the only three possibilities. Algebraically, let us try to understand the case of infinite solutions a little better. Why can’t a particular system have only two solutions, for instance? Let \(x_{1}\) and \(x_{2}\) be two solutions, then \((c+1)x_{1}-cx_{2}\) is also a solution. This is because:

\[ \begin{aligned} A[(c+1)x_{1}-cx_{2}] & =(c+1)Ax_{1}-cAx_{2}\\ & =(c+1)b-cb\\ & =b \end{aligned} \]

Since \(c\) is an arbitrary parameter, we see that there are infinitely many solutions.

Statement-2

Multiplying each equation by \(c\) corresponds to scaling each row of the matrix \(A\) by \(c\) and scaling each component of the vector \(b\) by \(c\). Multiplying each row of \(A\) by \(c\) is the same as multiplying the entire matrix by \(c\). Scaling each component of \(b\) by \(c\) is the same as scaling the vector \(b\) by \(c\). If \(x^{*}\) is a solution of the system \(Ax=b\), then:

\[ \begin{aligned} Ax^{*} & =b\\ cAx^{*} & =cb\\ (cA)x^{*} & =cb \end{aligned} \]

Note that the second step is valid only if \(c\) is a non-zero constant. We see that the new system is \((cA)x^{*}=cb\). The solution of this system is still \(x^{*}\). This shows that multiplying all equations by a non-zero constant doesn’t change the solution to the system.

Statement-3

Let the solution to \(Ax=b\) be \(x^{*}\). Then, \(Ax^{*}=b\).

\[ \begin{aligned} Ax^{*} & =b\\ cAx^{*} & =cb\\ cA\left(\frac{x^{*}}{c}\right) & =b \end{aligned} \]

We see that \(\frac{x^{*}}{c}\) is a solution to the system \(cAx=b\).

Statement-4

This is just a special case of the previous statement.