For a matric while doing a gaussian elimination if all elements in a particular columns below the pivot are zero and the pivot itself is zero then the matrix does not span the space

During gaussian elimination row echolon form of elimination is written as matrix multiplications the element of the row being subtracted from every other row in the main matrix below it is negative in the elimination matrix

inverse of an elimination matrix is in above case [1 0 0, 0 1 0, 0 2 1]