Question: Let's prove the theorem \((M N)^{T}=N^{T} M^{T}\).

Note: the following is a common technique for proving matrix identities.

1. Let \(M=\left(m_{j}^{i}\right)\) and let \(N=\left(n_{j}^{i}\right)\).
2. Multiply out \(MN\) and write out a few of its entries in the same form as in part (a). In terms of the entries of \(M\) and the entries of \(N\), what is the entry in row \(i\) and column \(j\) of \(MN\) ?
3. Take the transpose \((M N)^{T}\) and write out a few of its entries in the same form as in part (a). In terms of the entries of \(M\) and the entries of \(N\), what is the entry in row \(i\) and column \(j\) of \((M N)^{T}\) ?
4. Take the transposes \(N^{T}\) and \(M^{T}\) and write out a few of their entries in the same form as in part (a).
5. Multiply out \(N^{T} M^{T}\) and write out a few of its entries in the same form as in part a. In terms of the entries of \(M\) and the entries of \(N\), what is the entry in row \(i\) and column \(j\) of \(N^{T} M^{T} ?\)
6. Show that the answers you got in parts (c) and (e) are the same.

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