No invertible matrices

Show that there do not exist invertible matrices A, B \in \mathcal{M}_n \left( \mathbb{C} \right) such that A^2+B^2 = ( A + B )^2 and A^3+B^3 = ( A + B)^3.

Solution

Suppose, on the contrary, that such matrices do exist. Then

    \[(A + B) ^2 = A^2 +B^2 \implies AB + BA = \mathbb{O} \implies AB = - BA\]

and also

    \[(A+B)^3 = A^3 + B^3 \implies A^2 B = - B^2 A\]

Using the fact that AB =-BA we deduce that

    \begin{align*} -BA^2 &= A^2B \\ &= A \cdot A B\\ &= -A B \cdot A\\ &= BA \cdot A \\ &= BA^2 \end{align*}

The last means that BA^2 =\mathbb{O} which is impossible because both A and B are invertible ( and so must be the product ). Hence, the conclusion follows.

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