Elementary Matrix Operations#
Definition: Elementary Matrix Operations
The elementary row / column operations on a matrix \(A \in F^{m \times n}\) are the following:
- swapping two rows / columns;
- multiplying a row / column by \(\lambda \ne 0\);
- adding a row / column multiplied by \(\mu \in F\) to another row / column.
Theorem: Effects on Determinants
Elementary matrix operations have the following effects on the determinant of a square matrix \(A \in F^{n \times n}\):
- Swapping two rows / columns flips the sign of the determinant.
- Multiplying a row / column by \(\lambda \ne 0\) multiplies the determinant by \(\lambda\).
- Adding a row / column multiplied by \(\mu \in F\) to another row / column does not affect the determinant.
Proof
TODO
Theorem: Effects on Null Space
Elementary row operations preserve the null space of a matrix, but elementary column operations only preserve nullity.
Proof
TODO
Elementary Matrices#
Performing an elementary matrix operation on a matrix \(A \in F^{m \times n}\) can be represented as a multiplication of \(A\) with a square matrix:
- For an elementary row operation, perform the operation on the identity matrix \(I_m\) to obtain the square matrix \(E_m\). Applying the same elementary row operation to \(A\) is equivalent to the product \(E_mA\).
- For an elementary column operation, perform the operation on the identity matrix \(I_n\) to obtain the square matrix \(E_n\). Applying the same elementary column operation to \(A\) is the equivalent to the product \(AE_n\).