Final answer:
The properties of elementary matrices in linear algebra show that not all replacement matrices have a determinant of 1, and it is possible for a swap matrix and a scale matrix to have the same determinant (-1). An elementary matrix must be invertible and therefore cannot have a determinant of 0. A scaling elementary matrix has a determinant equal to the scaling factor, and any swap matrix has a determinant of -1. Generally, a swap matrix and a replacement matrix will not have the same determinant.
Step-by-step explanation:
We are addressing the properties of elementary matrices and their determinants in linear algebra. Elementary matrices represent simple linear transformations, such as row swapping (swap matrices), scaling (scale matrices), and adding multiples of rows to other rows (replacement matrices).
All replacement matrices have determinant 1. This statement is not always true. A replacement matrix may have a determinant other than 1 if it performs a row operation involving adding a multiple of one row to another.
It is impossible for a swap matrix and a scale matrix to have the same determinant. This statement is also not always true. A swap matrix always has a determinant of -1 because it represents a single row swap which changes the orientation of the matrix. Meanwhile, a scale matrix that scales a row by -1 would also have a determinant of -1.
There is an elementary matrix whose determinant is 0. False, an elementary matrix must be invertible, and a matrix with a determinant of 0 is not invertible.
The n × n elementary matrix realizing the scaling of a single row by a factor of α has determinant α. This is true, when an elementary matrix represents scaling a row by a factor of α, its determinant is indeed α.
The determinant of any swap matrix is -1. True, as previously mentioned, a swap matrix changes the orientation of the matrix, which results in a determinant of -1.
It is impossible for a swap matrix and a replacement matrix to have the same determinant. This is generally true since a replacement matrix usually has a determinant of 1 (assuming it doesn't scale a row), and a swap matrix has a determinant of -1.