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I'm attempting to prove that

$$ \left[ \begin{array}{c c} A & B \\ C & D \\ \end{array} \right]^\top = \left[ \begin{array}{c c} A^\top & C^\top \\ B^\top & D^\top \\ \end{array} \right]. $$

Intuitively, I can see that it's true. However, when I try to formally prove it, I quickly get lost in the indices. What tricks can I use to keep things straight?

Source: Exercise 2.6.16, P116, Intro to Linear Algebra, 4th Ed by Strang

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up vote 10 down vote accepted

Most people would just claim this is obvious and omit the proof, but if you don't want to do that then perhaps you could first prove that \begin{equation} \begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix} \end{equation} and \begin{equation} \begin{bmatrix} M \\ N \end{bmatrix}^T = \begin{bmatrix} M^T & N^T \end{bmatrix}. \end{equation} Then \begin{align} \begin{bmatrix} A & B \\ C & D \end{bmatrix}^T &= \begin{bmatrix} \begin{bmatrix} A \\ C \end{bmatrix}^T \\ \begin{bmatrix} B \\ D \end{bmatrix}^T \end{bmatrix} \\ &= \begin{bmatrix} A^T & C^T \\ B^T & D^T \end{bmatrix}. \end{align}

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