Question

how can you use to determinant of a matrix to determine whether the matrix has an inverse?

Answer 1

If the determinant is non-zero then the matrix is invertible.

Answer 2

Here is an intuitive way to remember this. You can write the inverse of a matrix \(A\) as \(\dfrac1{\det A}\) times the matrix of cofactors. If the determinant is zero, that will not be defined.