Question-48

Which of the following statements is/are true?

Note: \(\mathbb{R}\) denotes the set of all real numbers.

• There exist \(\mathbf{M} \in \mathbb{R}^{3 \times 3}, \mathbf{p} \in \mathbb{R}^{3}\) and \(\mathbf{q} \in \mathbb{R}^{3}\) such that \(\mathbf{Mx} = \mathbf{p}\) has a unique solution and \(\mathbf{Mx} = \mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{3\times 3} ,\mathbf{p} \in \mathbb{R}^{3}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{3}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has no solutions and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{2\times 3} ,\mathbf{p} \in \mathbb{R}^{2}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{2}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has a unique solution and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{3\times 2} ,\mathbf{p} \in \mathbb{R}^{3}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{3}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has a unique solution and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has no solutions.

Answer

• There exist \(\mathbf{M} \in \mathbb{R}^{3 \times 3}, \mathbf{p} \in \mathbb{R}^{3}\) and \(\mathbf{q} \in \mathbb{R}^{3}\) such that \(\mathbf{Mx} = \mathbf{p}\) has a unique solution and \(\mathbf{Mx} = \mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{3\times 3} ,\mathbf{p} \in \mathbb{R}^{3}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{3}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has no solutions and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{2\times 3} ,\mathbf{p} \in \mathbb{R}^{2}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{2}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has a unique solution and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has infinite solutions.
• There exist \(\displaystyle \mathbf{M} \in \mathbb{R}^{3\times 2} ,\mathbf{p} \in \mathbb{R}^{3}\) and \(\displaystyle \mathbf{q} \in \mathbb{R}^{3}\) such that \(\displaystyle \mathbf{Mx} =\mathbf{p}\) has a unique solution and \(\displaystyle \mathbf{Mx} =\mathbf{q}\) has no solutions.

Solution

The key to solving this problem is to note the following facts about a general system \(Ax = b\), where \(A \in \mathbb{R}^{m \times n}\) , \(b \in \mathbb{R}^m\):

• \(Ax = b\) has a solution if \(b\) is in the column space of \(A\).
• If \(Ax = b\) has a solution, it is unique if and only if the columns of \(A\) are linearly independent.
• The columns of \(A\) are linearly independent if and only if \(A\) has full column rank.

Thus we can divide the problem into three cases:

Case-1: \(m > n\)

If \(A\) has full column rank, we can always find:

• \(b_1\), some vector in the column space
• \(b_2\), some vector not in the column space

The first leads to a unique solution and the next gives no solution.

If \(A\) is rank deficient, we can always find:

• \(b_1\), some vector in the column space
• \(b_2\), some vector not in the column space

The first leads to infinite solutions and the next gives no solution.

Case-2: \(m = n\)

If \(A\) has full column rank, we are guaranteed a unique solution for any \(b\).

If \(A\) is rank deficient, the analysis is similar to the second point in case-1

Case-3: \(m < n\)

We can never get a unique solution as this system can never have full column rank. But we can still end up with a situation similar to the second point in case-1.