§3] 
MATRICES 
5 
Xi — <z I i£i+a 12 £ 2 + .... -\-CLip^p, 
A: 
Xp = ap I ^ l -\-ap 2 ^ 2 -\- .... -\-app%p 
except that the equations of the inverse A~ x are now 
more complicated (§ 3). 
3. Matrices. A linear transformation is fully defined 
by its coefficients, while it is immaterial what letters 
are used for the initial and the final variables. For 
example, when we wrote the equations for t~ I in § 2, 
we replaced the letters x, y which were first employed 
to designate the new variables by other letters X, Y. 
Hence the transformations t, r, and A in § 2 are fully 
determined by their matrices: 
the last having p rows with p elements in each row. 
Such a /»-rowed square matrix is an ordered set of p 2 
elements each occupying its proper position in the symbol 
of the matrix. The idea is the same as in the notation 
for a point (x, y) of a plane or for a point (x, y, z) in 
space, except that these one-rowed matrices are not 
square matrices. The matrix 
of the transformation t x = tr is called the product of the 
matrices m and ¡jl of the transformations t and r. Hence 
the element in the ith row and 7th column of the product 
of two matrices is the sum of the products of the succes