$A=\pmatrix{1&2&3&4&5\\2&3&4&5&6}.$ Find $\det(A^TA)$.

Question

Suppose $$A=\pmatrix{1&2&3&4&5\\2&3&4&5&6}$$ Find $\det(A^TA)$.

I know exactly how to calculate it by writing it as a $5\times5$ matrix. But how to calculate it smartly?

Answer 1

• Rank of $A$ is $2$, hence $A^TA$ cannot be rank $5$ and it must be singular.

• Hence the determinant must be $0$.