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

• MHB
• karush
In summary, a square matrix with a zero row is not invertible because the determinant will be zero, making it impossible to find an inverse matrix. This can be understood by looking at an example where the first row of the matrix consists entirely of zeros and considering the resulting matrix after multiplication with another matrix.
karush
Gold Member
MHB
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.

Last edited:
karush said:
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?

so what would AB look like in your example

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.

## 1. What does it mean for a matrix to have a zero row?

A zero row in a matrix means that all the elements in that row are equal to zero. This can also be referred to as a row of zeros.

## 2. Why is a square matrix with a zero row not invertible?

A square matrix with a zero row is not invertible because an invertible matrix must have a non-zero determinant. In a square matrix, the determinant is calculated by multiplying the elements in each row and column. If one of the rows is all zeros, the determinant will also be zero, making the matrix non-invertible.

## 3. Can a square matrix with a zero row have an inverse?

No, a square matrix with a zero row does not have an inverse. As mentioned before, an invertible matrix must have a non-zero determinant, which is not possible if one of the rows is all zeros.

## 4. How can you prove that a square matrix with a zero row is not invertible?

To prove that a square matrix with a zero row is not invertible, you can find the determinant of the matrix. If the determinant is equal to zero, then the matrix is not invertible. Additionally, you can also try to find the inverse of the matrix using different methods, and if you are unable to find an inverse, then the matrix is not invertible.

## 5. Are there any other conditions that make a square matrix not invertible?

Yes, apart from having a zero row, a square matrix can also be non-invertible if it has linearly dependent rows or columns, or if it has a determinant of zero. In general, a square matrix is not invertible if it is singular, meaning it does not have an inverse.

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