Which of the following statements is accurate about invertible matrices?

The statement regarding invertible matrices that is accurate is that the inverse always exists. An invertible matrix, also known as a non-singular matrix, has a crucial property: if a matrix is invertible, there is another matrix (its inverse) such that when both are multiplied together, they yield the identity matrix. This relationship is fundamental in linear algebra because it allows for the unique solution of linear systems represented by the equations associated with the matrix.

For a matrix to be invertible, it must be square, and its determinant must be non-zero. The existence of the inverse implies that the columns (and rows) of the matrix are linearly independent, allowing the system of equations to have a unique solution. If the determinant equals zero, the matrix is singular and thus not invertible.

While the other choices touch on concepts related to matrices, they do not accurately describe the properties of invertible matrices. For example, the determinant does not have to equal 1 for a matrix to be invertible; it simply cannot be zero. Additionally, a matrix with linearly dependent columns cannot be inverted because this dependence implies that the determinant is zero. Lastly, invertible matrices can indeed transform vectors, as they are often used to change the basis