The determinant of matrix $$A$$ is $$5$$ and the determinant of matrix $$B$$ is $$40.$$ The determinant of matrix $$AB$$ is _______.

The system of linear equations $$\left( {\matrix{ 2 & 1 & 3 \cr 3 & 0 & 1 \cr 1 & 2 & 5 \cr } } \right)\left( {\matrix{ a \cr b \cr c \cr } } \right) = \left( {\matrix{ 5 \cr { - 4} \cr {14} \cr } } \right)$$ has

The maximum value of the determinant among all $$2 \times 2$$ real symmetric matrices with trace $$14$$ is ______.

Let $$A$$ be an $$m\,\, \times \,\,n$$ matrix and $$B$$ an $$n\,\, \times \,\,m$$ matrix. It is given that determinant $$\left( {{{\rm I}_m} + AB} \right) = $$determinant $$\left( {{{\rm I}_n} + BA} \right),$$ where $${{{\rm I}_k}}$$ is the $$k \times k$$ identity matrix. Using the above property, the determinant of the matrix given below is $$\left[ {\matrix{ 2 & 1 & 1 & 1 \cr 1 & 2 & 1 & 1 \cr 1 & 1 & 2 & 1 \cr 1 & 1 & 1 & 2 \cr } } \right]$$