Question

The entries of a matrix are integers. Adding an integer to all entries on a row or on a column is called an operation. It is given that for infinitely many integers N one can obtain, after a finite number of operations, a table with all entries divisible by N. Prove that one can obtain, after a finite number of operations, the zero matrix.

• A. It is possible to obtain the zero matrix.
• B. It is not possible to obtain the zero matrix.
• C. The procedure will give identity matrix .
  
• D. The procedure will give Skew-Symmetric matrix.

Answer 1

The correct option is A It is possible to obtain the zero matrix.

Suppose the matrix has m rows and n columns and its entries are denoted by $a_{hk}$
Fix some a $j,1<j\le n,$
and consider an arbitrary $j,1<j\le m,$
The expression $E_{ij}=a_{11}+a_{ij}-a_{i1}-a_{1j}$ is an invariant for our operation. Indeed, adding k to the first row, it becomes $(a_{11}+k)+a_{ij}-a_{i1l}-(a_{1j}+k)=a_{11}+a_{i1}-a_{1j}.$
The same happens if we operate on the first column, the i th row or the j th column, while operating on any other row or column clearly does not change $E_{ij}$
is divisible by infinitely positive integers N, hence $E_{ij}=0$
We deduce that $a_{11}-a_{1j}=a_{ij}=c,$
For all $i,1<i\le m.$
Adding c to all entries in column j will make this column identical to the first one. In the same way we can make all columns identical to the first one. Now, it is not difficult to see that operating on rows we can obtain the zero matrix.