Q: Find a $3\times5$ full rank matrix for which $\det(A^TA)\ne0$.
I know that if $\det(A^TA)\ne0$ then the columns of $A$ need to be linearly independent, but I cannot find a proper matrix. Any matrix I test has linearly dependent columns. Any suggestions?
If $A$ has 3 lines and 5 columns, then $\det A^T A = 0$. So it is impossible to find a matrix that satisfies your requirements.
On the other hand, if $A$ has 5 lines and 3 columns, this is pretty easy, you can take the identity matrix $I_3$ for the first three lines, and zero for the last two lines. Then $\det A^T A = 1$.