What does it mean to have a determinant equal to zero? After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$.
I hope someone can explain this to me in plain English.

For an $n\times n$ matrix, each of the following is equivalent to the condition of the matrix having determinant $0$:


*

*The columns of the matrix are dependent vectors in $\mathbb R^n$

*The rows of the matrix are dependent vectors in $\mathbb R^n$

*The matrix is not invertible. 

*The volume of the parallelepiped determined by the column vectors of the matrix is $0$.

*The volume of the parallelepiped determined by the row vectors of the matrix is $0$.

*The system of homogenous linear equations represented by the matrix has a non-trivial solution. 

*The determinant of the linear transformation determined by the matrix is $0$. 

*The free coefficient in the characteristic polynomial of the matrix is $0$. 
Depending on the definition of the determinant you saw, proving each equivalence can be more or less hard.
