The statement is false.

If $A$ is an invertible square matrix, then it cannot have a column (or a row) of zeros. This is because for a matrix to be invertible, its determinant must be non-zero. If any column (or row) of the matrix is all zeros, the determinant of the matrix will be zero, and thus the matrix will not be invertible.

Why this is true:

• The determinant of a matrix is calculated based on the values of its elements. If a matrix has an entire column of zeros, it means the matrix is singular (its rows or columns are linearly dependent), and its determinant will be zero. This directly implies that the matrix is not invertible.

Let me know if you need further clarification on this!

Related Questions:

1. What is the definition of an invertible matrix?
2. How can you determine if a matrix is singular?
3. What happens to the determinant if two rows or columns are identical?
4. How do row operations affect the invertibility of a matrix?
5. What is the relationship between the rank of a matrix and its invertibility?

Tip:

To test the invertibility of a matrix, compute its determinant. If the determinant is zero, the matrix is not invertible; otherwise, it is.