Question

Mark each statement True or False. Justify each answer.
a. A homogeneous equation is always consistent.
A. True. A homogenous equation can be written in the form $A x=0$, where $A$ is an $m \times n$ matrix and 0 is the zero vector in $R^m$. Such a system $A x=0$ always has at least one nontrivial solution. Thus a homogenous equation is always consistent.

B. False. A homogenous equation can be written in the form $\mathrm{Ax}=\mathbf{0}$, where $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathbf{0}$ is the zero vector in $\mathbb{R}^m$. Such a system $\mathrm{Ax}=\mathbf{0}$ always has at least one solution, namely, $\mathbf{x}=0$. Thus a homogenous equation is always inconsistent.

C. True. A homogenous equation can be written in the form $A \mathbf{x}=\mathbf{0}$, where $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathbf{0}$ is the zero vector in $\mathbb{R}^m$. Such a system $\mathrm{Ax}=\mathbf{0}$ always has at least one solution, namely, $x=0$. Thus a homogenous equation is always consistent.

D. False. A homogenous equation can be written in the form $A \mathbf{x}=\mathbf{0}$, where $\mathrm{A}$ is an $\mathrm{m} \times \mathrm{n}$ matrix and $\mathbf{0}$ is the zero vector in $\mathbb{R}^m$. Such a system $A \mathbf{x}=\mathbf{0}$ always has at least one nontrivial solution. Thus a homogenous equation is always inconsistent.