Suppose A is a n x k matrix and you have $\displaystyle A(A^TA)^{-1}A^T$. All that is equivalent to $\displaystyle I$, 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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- Oct 23rd 2009, 03:06 AMgarymarkhovFiguring out rows and columns after matrix multiplication
Suppose A is a n x k matrix and you have $\displaystyle A(A^TA)^{-1}A^T$. All that is equivalent to $\displaystyle I$, but how many rows does I have? Is it n or k, and what is a quick way to figure it out?

- Oct 23rd 2009, 05:18 AMHallsofIvy
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 $\displaystyle A^T$ has k columns and n rows. In order to be able to multiply $\displaystyle (A^TA)^{-1}A^T$, $\displaystyle (A^TA)^{-1}$ must have n columns and, since it is invertible it is square, n rows. That means that $\displaystyle (A^TA)^{-1}A^T$ has k columns and n rows. And then $\displaystyle A(A^TA)^{-1}A^T$ 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.