Question

Question #f4e98

Answer 1

`tr( 2A^7+3A^5+4A^2+A+I ) = 6`

Explanation:

If `A` is nilpotent with order `2`; then

`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`

Answer 2

If `A^2=0` then

`B=2A^7+3A^5+4A^2+A+I=A+I`

and `"tr"(B)="tr"(A)+"tr"(I)=3+3=6`