Final answer:
Every elementary matrix is invertible because each one represents a reversible elementary row operation. These operations can be undone by their respective inverses, assuring that all elementary matrices possess inverse matrices.
Step-by-step explanation:
The assertion that every elementary matrix is invertible is indeed true. Elementary matrices stem from performing elementary row operations on the identity matrix. Each of these operations—swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another—is reversible, hence preserving invertibility.
An invertible matrix is one that has an inverse, such that when multiplied together, the product is the identity matrix. Conversely, the inverse of an invertible matrix will undo the operation performed by the original matrix, taking you back to the identity matrix as well. Elementary matrices can generate any invertible matrix by a finite sequence of elementary row operations, which reaffirms their invertibility.
As for the options given:
- Option A is false because it suggests elementary matrices can be non-invertible, which is incorrect.
- Option B is the correct choice, linking invertibility to the reversibility of row operations.
- Option C misinterprets the extent of row operations, assuming that failure to reach the identity matrix implies non-invertibility of the operations, which is incorrect.
- Option D, while indirectly true, posits a subtle logical fallacy: not every product of elementary matrices is invertible, but each of the elementary matrices themselves is.
Thus, the final answer is: True; since each elementary matrix corresponds to a row operation, and every row operation is reversible, every elementary matrix has an inverse matrix.