I saw a phrase in my book that says " it's impossible to convert a matrix with a last row all zeros into an identity matrix ".
But I was curious if there is any combination of elementary operations to convert it or a proof for the above statement.
So can someone help me?
Best Answer
An identity matrix is a square matrix with $1$'s all along the diagonal. Thus, an identity matrix has a $1$ in every column (and every row).
There are three elementary row operations. None of them are capable of taking a matrix that has all zeros in a given column and transforming it into a matrix with a non-zero element in that column.
Swap Rows
If you swap two rows that both have zeros in the $n$th column, those rows will continue to have zeros in the $n$th column.
Scalar Multiplication
It is impossible to multiply a row with a zero in the $n$th column by a scalar to cause it to have a non-zero element in the $n$th column, since there is no scalar by which you can multiply $0$ to get anything other than $0$.
Row Addition
If multiple rows have $0$'s in the $n$th column, it's impossible to add them together and get a row that has a non-zero element in the $n$th column. That would require adding zero to itself to get something other than $0$, which is impossible.
Since it's impossible to create a non-zero element in the $n$th row of a matrix if the $n$th row contains all zeros, it is impossible to convert said matrix into an identity matrix, which requires a non-zero element ($1$) in every column.