Competition problem — Algebra — Olympiad Maths Prep

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Algebra Difficulty 5.1 AIME, harder Prove it Romania

Consider matrices $A$ , $B$ , $C$ , $D \in \mathcal{M}_n(\mathbb{C})$ , $n \ge 2$ și $k \in \mathbb{R}$ so that $AC + kBD = I_n$ and $AD = BC$ . Demonstrate that $CA + kDB = I_n$ and $DA = CB$ .

Solution

$(CA + kDB) - w(DA - CB) = I_n$
and
$(CA + kDB) + w(DA - CB) = I_n,$
which give the result. For $k = 0$ we get $AC = I_n$ , so $CA = I_n$ . From $AD = BC$ and $CA = I_n$ we obtain $ADA = B$ and $DA = CB$ .

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