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.

If the determinant of a square matrix $n\times n$ $A$ is zero, then $A$ is not invertible. This is a crucial test that helps determine whether a square matrix is invertible, i.e., if the matrix has an inverse. When it does have an inverse, it allows us to find a unique solution, e.g., to the equation $Ax = b$ given some vector $b$.
When the determinant of a matrix is zero, the system of equations associated with it is linearly dependent; that is, if the determinant of a matrix is zero, at least one row of such a matrix is a scalar multiple of another.
[When the determinant of a matrix is nonzero, the linear system it represents is linearly independent.]
When the determinant of a matrix is zero, its rows are linearly dependent vectors, and its columns are linearly dependent vectors.
