the answer of this question is $n<3$ , why? I mean how we decided that is $n<3$ when $A$ multiply $A^\top$ will be singular?
2026-03-26 19:01:49.1774551709
suppose $A$ is an $m\times n$ real matrix and $A^\top A$ is $3\times3$ nonsingular matrix, then $n =$?
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How the resulting matrix of $A^T A$ corresponds to $m \times n$ should be quite visible when familiarizing yourself with the matrix products as stated in the comments.
As to why $n \geq 3$, we have to look at the rank of the matrix. If $n < 3$, the rank of the matrix would be less than $3$ as well (since rank is equal to column rank).
Now furthermore the rank of $A^T$ equals the rank of $A$ (again rank equals column rank equals row rank).
And at last if the rank of $A$ was less than $3$, the rank of $A^T A$ (the composition of $A^T$ and $A$) would have to be less than or equal to the minimum of the ranks of the two matrices. Which would be less than 3 and $A^T A$ would not be invertible/non-singular.
As such we get that $n \geq 3$.