Question 45: Compute \(\det {A^T}A\) and \(\det A{A^T}\) for several random \(4 \times 5\) matrices and several random \(5 \times 6\) matrices. What can you say about \({A^T}A\) and \(A{A^T}\) when \(A\) has more columns than rows?

Suppose \(A\)is a matrix with more columns than rows.Then, the columns of \(A\) must be linearly dependent. Also, the equation \(Ax = 0\) must have a non-trivial solution \(x\). Hence,

\(\begin{array}{c}\left( {{A^T}A} \right){\mathop{\rm x}\nolimits} = {A^T}\left( {A{\mathop{\rm x}\nolimits} } \right)\\ = {A^T}0\\ = 0,\end{array}\)

and the equation \(\left( {{A^T}A} \right){\mathop{\rm x}\nolimits} = 0\) has a non-trivial solution.

According to the invertible matrix theorem,\({A^T}A\) is not invertible because it is a square matrix. This argument does not apply in general to \({A^T}A\) since \({A^T}\)has more rows than columns, so its columns are not always linearly dependent.

Thus,\({A^T}A\) is not invertible,and the equation \(\left( {{A^T}A} \right){\mathop{\rm x}\nolimits} = 0\) has a non-trivial solution.