Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 12 June 2011


Let m and n be positive integers. Let R be a ring.


( 2 3 1 2 ) + ( 5 2 1 6 ) = ( 7 5 2 8 ) , ( 2 3 1 2 ) ( 5 2 1 6 ) = ( 13 22 7 14 ) and ( 2 3 1 2 ) t = ( 2 1 3 2 ) .

HW: Show that if a,b,c are n×n matrices then (ab)c =a(bc), a(b+c) = ab+ac and (b+c)a =ba+ca.

HW: Give an example of n×n matrices a and b such that abba.

Let S be a set.

Let m and n be positive integers and let R be a ring.
(a)   The set
Mm×n (R)= {m×n matrices with entries in R} ,
with operation addition is an abelian group with zero element 0 Mm×n (R) given by
0ij =0 for i=1,2,,m, and j=1,2,,n.
(b)   The set Mm×n (R) with operations addition and scalar multiplication is an R-module.
(c)   The set Mn(R) = Mn×n (R) with operations addition and product is a ring with identity 1 Mn(R) given by
1ij = δij for i=1,2,,n, and j=1,2,,n.

HW: Show that Mn×1 (R) Rn as R-modules.


(a)   The set of triangular matrices and the set of diagonal matrices are subrings of Mn(R).
(b)   The ideals of Mn(R) are Mn(I), where I is an ideal of R.

Let a be an n×n matrix with entries aij.

Let 𝔽 be a field and let n >0.
(a)   Up to constant multiples, tr: Mn(𝔽) 𝔽 is the unique function such that
(a1)   If c𝔽 and a,b Mn(𝔽) then
tr(a+b) =tr(a) +tr(b) and tr(ca)= ctr(a).
(a2)   If a,b Mn(𝔽) then
tr(ab) = tr(ba) .
(b)   Identify Mn(𝔽) with the 𝔽-module 𝔽n × × 𝔽n ntimes ,
Mn(𝔽) 𝔽n × × 𝔽n ntimes a (a1| a2|| an) ,      where ai are the columns of a.
The function det: Mn(𝔽)𝔽 is the unique function such that
(b1)   (columnwise linear) If i {1,2,,n} and c𝔽 then
det (a1| | ai+bi || an) = det (a1| | ai|| an) + det (a1| | bi|| an)
det (a1| | cai|| an) = c det (a1| | ai|| an) ,
(b2) If i {1,2,, n-1} then
det (a1| | ai| ai+1| | an) = - det (a1| | ai+1| ai| | an) ,
(b3)   if a,b Mn(𝔽) then
det(ab) = det(a) det(b) .

Notes and References

A matrix is just a table of numbers, and hence matrices appear everywhere. These notes are taken from notes of Arun Ram from 1999. A nice solid reference is [HP].


[HP] W.V.D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I. Reprint of the 1947 original. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1994. viii+440 pp. ISBN: 0-521-46900-7, 14-01 (01A75) Methods of algebraic geometry I, Cambrdige University Press, MR1288305.

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