How to prove the equation about the determinant of Matrix $M$, i.e.,
$|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$
where $a$, $b$ and $c$ are arbitrary vectors.
This euqation is encountered in An introduction to continuum mechanics P.55 , authored by G.N. Reddy. It's subsequently used to prove that the determinant of Deformation Gradient is the change of volume during deformation.
I would be greatful if somebody could shed some lights on it.
For all vectors $a, b, c \in{\mathbb R}^3$, one has $(a\times b)\cdot c = \det(a,b,c)$ by Laplace expansion. Hence \begin{equation} (M a \times M b)\cdot M c = \det(M a, M b, M c) = \det(M[a, b, c]) = \det(M)\det(a, b, c) = \det(M) (a\times b)\cdot c \end{equation}