Effects of Row and Column Operations on Determinants

Theorem: Effects of Row and Column Operations on Determinants

For any square matrix $A\in F^{n\times n}$:

• Swapping two rows or two columns changes the algebraic sign of $A$‘s determinant.

• Multiplying a single row or a single column by $\lambda \in F$ results in $A$‘s determinant being multiplied by $\lambda$.

• Adding a non-zero multiple of one row or column to another row or column has no effect on $A$‘s determinant.

PROOF

TODO