After looking in my book for a couple of hours, I’m still confused about what it means for a (n×n)matrix A to have a determinant equal to zero, det.
I hope someone can explain this to me in plain English.
Answer
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 nontrivial 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.
Attribution
Source : Link , Question Author : user2171775 , Answer Author : Ittay Weiss