MHB 5.3 Show that a square matrix with a zero row is not invertible.

[karush]

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.

 

[Evgeny.Makarov]

if
$$\det \begin{pmatrix}1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}=1$$
then $$\begin{pmatrix}
1&0&0\\ 0&1&0\\ 3&0&1\end{pmatrix}^{-1}
=\begin{pmatrix}1&0&0\\ 0&1&0\\ -3&0&1
\end{pmatrix}$$

This reminds me a joke. A tourist asks a local resident: "If I go down this street, will there be a railway station?" The local replies, "The station will be there even if you don't go down that street".

Suppose the $i$th row of a square matrix $A$ consists entirely of zeros. What is the $i$th row in the matrix $AB$ for any size-compatible matrix $B$? Can $AB$ be the identity matrix?

 

[karush]

so what would AB look like in your example

 

[Evgeny.Makarov]

One cannot say much about about $AB$ in general without knowing more about $A$ and $B$, but we can know the $i$th row of $AB$. I suggest you use the definition of matrix multiplication to find what that row is. If you want, you can consider an example where $A$ is the matrix from your original post where the first row has all zeros and $B$ is an arbitrary 3x3 matrix. However, it is pretty obvious what the $i$th row of $AB$ is in general if the $i$th row of $A$ consists of zeros.