Sparse Matrix transformation to locate columns containing zeros

[taknevski]

Suppose I have a MxN matrix and each column may contain exactly 1 zero or
no zeros. Is there a transformation that when applied to this matrix will
return another 1xN matrix with a 0 in the corresponding column if that
column in the original matrix contains a 0 or a 1 in the column position if
that column in the original matrix contains no zeros.

For example if the 4x4 original matrix is

1 1 8 9
2 0 7 2
3 2 9 6
4 5 0 2

is there a transformation that will give me

1 0 0 1

since the 2nd and 3rd column contain a zero and the 1st and 4th columns
contain no zeros?

 

[fresh_42]

No. Your matrix has full rank and is invertible. You cannot reduce it to one row. But as it is invertible, you can solve
$$
\begin{bmatrix}1&1&8&9\\2&0&7&2\\3&2&9&6\\4&5&0&2\end{bmatrix}\cdot \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}= \begin{bmatrix}1\\0\\0\\1 \end{bmatrix}
$$