I am asking this question in the context of the following question:
Need help to understand a solution to a polynomial problem.
In the above question, it was stated that the characteristic polynomial is
$$\lambda^2-(a^3+d^3+3abdc+3bcd)\lambda+(ad-bc)^3=0$$
for the matrix
$$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^3$$
Since assumingly it is given by the solution of a textbook, the calculation should be somehow reasonable.
When I multiply out the matrix three times I get
$$\begin{pmatrix} a^3+2abc+bcd & a^2b+b^2c+abd+bd^2 \\ a^2c+acd+bc^2+cd^2 & abc+2bcd+d^3 \end{pmatrix}$$
I can see how the $\lambda$ term has coefficient $a^3+d^3+3abc+3bcd$ even though the calculation is not so easy, but it really seems impossible to do the $(ad-bc)^3$ part by hand.
There must be something missing instead of doing the $4$-term by $4$-term multiplication by hand.
