# [SOLVED]5.3 Show that a square matrix with a zero row is not invertible.

#### karush

##### Well-known member
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.

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#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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

##### Well-known member
so what would AB look like in your example

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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.