The minimum eigenvalue of the following matrix is $$\left[ {\matrix{ 3 & 5 & 2 \cr 5 & {12} & 7 \cr 2 & 7 & 5 \cr } } \right]$$

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 $$\left[ {\matrix{ 2 & 1 & 1 & 1 \cr 1 & 2 & 1 & 1 \cr 1 & 1 & 2 & 1 \cr 1 & 1 & 1 & 2 \cr } } \right]$$

Consider a vector field $$\overrightarrow A \left( {\overrightarrow r } \right).$$ The closed loop line integral $$\oint {\overrightarrow A \bullet \overrightarrow {dl} } $$ can be expressed as

The divergence of the vector field $$\,\overrightarrow A = x\widehat a{}_x + y\widehat a{}_y + z\widehat a{}_z\,\,$$ is