# 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.