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Let $A, B, C, D$ will be square matrices $n \times n$ over $R$.
Suppose that with $AB ^{T}$ and $CD ^{T}$ are symmetrical and $AD ^{T} - BC ^{T} = I$.
Show that: $A ^{T} D - C ^{T} B = I$:

I'm trying to transform the last equation, but it comes to me something like this:
$A^{T} C = C ^{T} A$
I do not know I can prove it. maybe I do it not as it should be, I would ask for any hint.

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    what does `polish student` have to do with anything? I may be dense in asking such a question. – 2017-01-04
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    Keep the english simple if replying, I should think. – 2017-01-04
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    @Paul ah thank you, we are doing that now? maybe a tag will be required at some point knowing how things go on MSE! – 2017-01-04
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    Putnam 1986-B6. – 2017-01-04
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    Here's a hint taken from the relevant MAA book http://imgur.com/poEbE21 – 2017-01-04

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Catalin Zara has effectively answered this. Solution here, B6.