Final answer:
Statement 4, which states that the matrix is invertible, is not equivalent to the others because it contradicts the implications that the matrix has a determinant of zero or is singular, both of which mean the matrix is not invertible.
Step-by-step explanation:
The question asks which of the given statements about a matrix with four rows is not equivalent to the others. To determine this, we must understand what each statement implies regarding the matrix.
- Statement 1: The determinant of the matrix is zero.
- Statement 2: The matrix has a row of zeros.
- Statement 3: The matrix is singular.
- Statement 4: The matrix is invertible.
Statement 1 implies that the matrix does not have an inverse; a matrix with a determinant of zero is not invertible. This is equivalent to Statement 3, as a singular matrix is another way of saying that the matrix does not have an inverse. Statement 2 often leads to Statement 1 because if a matrix has a row of zeros, the determinant is usually zero. This leaves us with Statement 4, which asserts that the matrix is invertible. This is the opposite of what the other statements imply, making it the statement that is not equivalent to the others. In the context of matrices, to be invertible, a matrix must have a non-zero determinant and must not be singular. Therefore, Statement 4 is the one that is not equivalent.