$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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proof

Conditions for a square matrix having a unique solution

If $A$ is square, then $Ax=0$ has a unique solution $(x= 0)$ if and only if $A$ is row equivalent to the identity matrix.

Concepts

Coming soon

If

We require that $A$ is row equivalent to the identity matrix. This would mean $Ax=0$ has the same solution as $I_n x=0$, where $I_n$ is the $n\times n$ identity matrix.

Only If

Coming soon

Proof

We use the fact that two row equivalent matrices have the same solution, and only the identity matrix has a unique solution for $Ax=0$.

Conditions for a square matrix having a unique solution

If $A$ is square, then $Ax=0$ has a unique solution $(x= 0)$ if and only if $A$ is row equivalent to the identity matrix.

Concepts

Coming soon

If

We require that $A$ is row equivalent to the identity matrix. This would mean $Ax=0$ has the same solution as $I_n x=0$, where $I_n$ is the $n\times n$ identity matrix.

Only If

Coming soon

Proof

We use the fact that two row equivalent matrices have the same solution, and only the identity matrix has a unique solution for $Ax=0$.
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Conditions for a square matrix having a unique solution

If $A$ is square, then $Ax=0$ has a unique solution $(x= 0)$ if and only if $A$ is row equivalent to the identity matrix.

Concepts

Coming soon

If

We require that $A$ is row equivalent to the identity matrix. This would mean $Ax=0$ has the same solution as $I_n x=0$, where $I_n$ is the $n\times n$ identity matrix.

Only If

Coming soon

Proof

We use the fact that two row equivalent matrices have the same solution, and only the identity matrix has a unique solution for $Ax=0$.

Conditions for a square matrix having a unique solution

If $A$ is square, then $Ax=0$ has a unique solution $(x= 0)$ if and only if $A$ is row equivalent to the identity matrix.

Concepts

Coming soon

If

We require that $A$ is row equivalent to the identity matrix. This would mean $Ax=0$ has the same solution as $I_n x=0$, where $I_n$ is the $n\times n$ identity matrix.

Only If

Coming soon

Proof

We use the fact that two row equivalent matrices have the same solution, and only the identity matrix has a unique solution for $Ax=0$.
FullPage
result
concepts
hypothesis
implications
proof