Linear Systems and Matrix Decompositions - Linear Algebra - Data Science & Artificial Intelligence Previous Year Questions

A

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.

B

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 no solutions and \(\mathbf{Mx} = \mathbf{q}\) has infinite solutions.

C

There exist \(\mathbf{M} \in \mathbb{R}^{2 \times 3}, \mathbf{p} \in \mathbb{R}^2\), and \(\mathbf{q} \in \mathbb{R}^2\) such that \(\mathbf{Mx} = \mathbf{p}\) has a unique solution and \(\mathbf{Mx} = \mathbf{q}\) has infinite solutions.

D

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

A

\(O(n)\)

B

\(O(n^2)\)

C

\(O(n^3)\)

D

\(O(n^4)\)