INDETEEMIXATE EQUATIOI^-S Ax\D CONGEUEJ^CES. 
^1, 25 
^1, 35 • 
■ • ^1, n 
0 5 ^2, 25 
^2, 35 • 
•• ^2,« 
:! 0 5 ^3,25 
^3, 35 • 
• ^3, n 
^ 5 25 
35 • 
h 1 
W’ll6r6 jXii f*'!, u ^ 1 , n 25 ^n, !• 
If this matrix be post-multiplied by the unit, 
I 1; ^25 ^35 • • • 
I 0, 1 , 0, ... 0 
I 0 , 0 , 1,...0 
0, 0, 0,...l 
>1 
)lO 
(65.) 
the constituents ^ will be changed into while all the other constituents will 
remain unaltered ; so that by assigning proper values to the numbers we may 
bring the given matrix |j«|| into the form 
f^i, 
^' 1 , 25 
35 • 
• ^' 1 , n 
0, 
^2, 25 
^2, 35 • 
• ^2, n 
0 , 
^3, 25 
^3, 35 • 
• • ^3, n 
O 5 
^h,25 
35 • 
• ^n,n 
where r, ,- verifies the inequality 
From this it is easy to infer that if a matrix of the type {n—l)x{'n—l) can be reduced 
to the form (62.), the same reduction is possible for a matrix of the type nxn,!. e. since 
that reduction is possible when w = l, n = .. it is possible for every value of n. 
To prove that jlalj is equivalent (by post-multiplication) to only one of the reduced 
matrices (62.), it is sufficient to show that no two reduced matrices can be equivalent. If 
yx\ and la'\\ be two reduced matrices, and Ijvjj a unit-matrix, such that ||«|| x ||'y||=j|«'||, it may 
be infen-ed, by comparing the corresponding constituents of the two matrices ||«|| X ||v|l and 
\a] (beginning with the lowest horizontal rows of each and proceeding upwards), that 
all the constituents of [jy] which lie below its principal diameter are zero ; and conse- 
quently that the constituents of the principal diameter itself are all positive units. 
Further, that the constituents above the principal diameter of ijy|| are likewise zero, may 
be established (for each line of constituents parallel to the diameter, beginning with 
that nearest to it) by means of the inequalities (63.) which are satisfied by the consti- 
tuents both of |j«|j and It thus appears that two reduced matrices cannot be equi- 
valent, without being identical. It will be observed that the reducing unit is unique ; 
