You’ve provided three different matrices $M_3$, $M_4$, and $M_5$ which represent specific elementary row and column operations.

1. Matrix $M_3$ represents adding row 3 to row 1: 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ This operation corresponds to the elementary matrix for adding row 3 to row 1, leaving other rows unchanged.
2. Matrix $M_4$ represents swapping columns 1 and 4: 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}$$ This swaps columns 1 and 4, which is a permutation matrix for column swapping.
3. Matrix $M_5$ represents subtracting row 2 from each of the other rows: 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{pmatrix}$$ This matrix applies the operation of subtracting row 2 from rows 1, 3, and 4.

Would you like further clarification or details on how these matrices operate?

Here are some related questions to deepen your understanding:

1. How can elementary matrices be used to perform row reduction?
2. What are the implications of using an elementary matrix to swap rows?
3. How does matrix multiplication affect the original matrix when using an elementary matrix?
4. What is the geometric interpretation of these matrix operations?
5. How would you undo a row or column operation applied by an elementary matrix?

Tip: Elementary matrices can be easily inverted. The inverse of an elementary matrix corresponds to undoing the elementary row or column operation.