Matrices- Trace

Hi amm345,

Quote:

Originally Posted by amm345

show that for all A,B nxn matrices over the field F:
a)trace(X+Y)=trace(X) + trace(Y)
b)trace(XY)=trace(YX)
c)show that there are no nxn matrices X,Y such that XY-YX= indentity martrix

a) follows from the definition of matrix addition, $tr(A+B) = \sum_{i=1}^n a_{ii}+b_{ii} = \sum_{i=1}^n a_{ii}+\sum_{i=1}^n b_{ii} = tr(A)+tr(B)$

b) uses commutativity of the field, $tr(AB)= \sum_{i=1}^n \sum_{j=1}^n a_{ij} b_{ji} =\sum_{j=1}^n \sum_{i=1}^n b_{ji}a_{ij}= tr(BA)$

c) follows from b)