Show that matrix-multiplication is associative. matrix element of matrix multiplication ABC:
(( AB)C )ij i | ( AB)C | j i | ( AIB)IC | j k ,k i | A k k B k k C | j ( aik bkk ) ck j aik ( bkk ck j ) [I.1] Alternate answer:
uses the same idea,
( A( BC))ij k Aik
m
BkmCmj m
k
Aik B km Cmj (( AB)C) ij Aik ( BkmCmj ) ( Aik Bkm ) C mj
[I.2]
Show that the product of two orthogonal matrices is also orthogonal. 1
T
An orthogonal matrix is characterized by A A ; consider two such matrices A, B, and see if we get the inverse when we transpose,
( AB)T BT AT B 1 A1 ( AB) 1 Alternate answer:
[I.3]
consider the matrix elements,
T bsi ( A A) sr brj ( AB)T AB ( AB)ik T ( AB) kj ( AB) ki ( AB) kj aks bsi akr brj bsi aks akr brj bsi ask T akr k r brj T
T
b si srbrj bri brj (bir ) brj ( B B )ij ij ( I)ij
Hm. That was kind of neat. I learned how to manipulate indices .
