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#### karush

##### Well-known member

- Jan 31, 2012

- 2,687

Show that a square matrix with a zero row is not invertible.

first a matrix has to be a square to be invertable

if

$$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$

then $$\begin{pmatrix}

1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1}

=\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1

\end{pmatrix}$$

but if $r_1$ is all zeros

$$\det \begin{pmatrix}

0&0&0\\ 0&1&0\\ 3&0&0

\end{pmatrix}=0$$

then

$$\begin{pmatrix}

0&0&0\\ 0&1&0\\ 3&0&0

\end{pmatrix}^{-1} DNE$$

ok I,m not real sure formally why this is ...

I could only do so with an example.

first a matrix has to be a square to be invertable

if

$$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$

then $$\begin{pmatrix}

1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1}

=\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1

\end{pmatrix}$$

but if $r_1$ is all zeros

$$\det \begin{pmatrix}

0&0&0\\ 0&1&0\\ 3&0&0

\end{pmatrix}=0$$

then

$$\begin{pmatrix}

0&0&0\\ 0&1&0\\ 3&0&0

\end{pmatrix}^{-1} DNE$$

ok I,m not real sure formally why this is ...

I could only do so with an example.

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