EQUIVALENT MATRICES 
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(as right factor). To find hud and dbk we have only 
to interchange the words first and second in what pre 
cedes. 
The product cd (or dc) may be obtained from d by 
interchanging the two rows (or columns) of d. 
The product e u d may be obtained from d by inserting 
the factor u before each element of the first row of d. 
The product de u may be obtained from d by inserting 
the factor u after each element of the first column of d. 
Hence for any n, matrix d is equivalent to those 
and only those matrices which may be derived from it 
by any succession of the following elementary transforma 
tions: 
i) The addition to the elements of any row of the 
products of any element k of the set S into the corre 
sponding elements of another row, k being used as a 
left factor, 
ii) The addition to the elements of any column of the 
products of the corresponding elements of another 
column into any element k of S, k being used as a right 
factor. 
iii) The interchange of any two rows or of two 
columns. 
iv) The insertion of the same unit factor before each 
element of any row. 
v) The insertion of the same unit factor after each 
element of any column. 
We shall call the element d IX of matrix d its first 
element. If d^o there exists by (iii) an equivalent 
matrix whose first element is not zero. 
Lemma i. If the first element of a matrix d is not 
zero and is a left divisor of every element of the first row and