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Math Help - Figuring out rows and columns after matrix multiplication

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    Member garymarkhov's Avatar
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    Figuring out rows and columns after matrix multiplication

    Suppose A is a n x k matrix and you have A(A^TA)^{-1}A^T. All that is equivalent to I, but how many rows does I have? Is it n or k, and what is a quick way to figure it out?
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    Quote Originally Posted by garymarkhov View Post
    Suppose A is a n x k matrix and you have A(A^TA)^{-1}A^T. All that is equivalent to I, but how many rows does I have? Is it n or k, and what is a quick way to figure it out?
    If matrix X has m columns and n rows and matrix Y has p columns and q rows, in order to be able to multiply XY, we must have that m= q and then XY has p columns and n rows.

    If A has n columns and k rows then A^T has k columns and n rows. In order to be able to multiply (A^TA)^{-1}A^T, (A^TA)^{-1} must have n columns and, since it is invertible it is square, n rows. That means that (A^TA)^{-1}A^T has k columns and n rows. And then A(A^TA)^{-1}A^T is the product of a matrix with n columns and k rows by a matrix with k columns and n rows. k= k and the product is a square matrix with n rows and columns.
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