Question #f4e98
2 Answers
Mar 15, 2017
# tr( 2A^7+3A^5+4A^2+A+I ) = 6 #
Explanation:
If
# A^2 = 0 #
And more importantly;
# A^m = 0 AA m in NN, m ge 2#
From which we can deduce that:
# tr(A^m) = 0 AA m in NN, m ge 2#
And we can use the trace properties:
# tr(A+B) = tr(A)+tr(B) #
# tr(mA) \ \ \ \ = m \ tr(A) #
# tr(I_n) \ \ \ \ \ \ \ = n # where#I_n# is the#nxxn# identity matrix
And so:
# tr( 2A^7+3A^5+4A^2+A+I ) #
# " " = tr( 2A^7) +tr(3A^5)+tr(4A^2)+tr(A)+tr(I) #
# " " = 2tr( A^7) +3tr(A^5)+4tr(A^2)+tr(A)+tr(I) #
# " " = 0 +0+0+tr(A)+tr(I) #
# " " = tr(A)+tr(I) #
# " " = 3+3 #
# " " = 6 #
Mar 15, 2017
If
and
