- The determinant of any identity matrix is always 1.
- If you switch the rows or columns of a matrix, its determinant stays the same.
- If you multiply a row (or column) of a matrix (call it A) by a number k, then the determinant of that matrix is k*det(A). For example, if
$ A=\begin{bmatrix} 1 & 2 & 3\\ 2 & 3 & 5\\ 3 & 5 & 0\end{bmatrix} $
then det(A)=8. Thus you can conclude if you multiply the top row by 10 (call the resulting matrix B), then det(B)=10*det(A)=80.
$ \text{det}\begin{pmatrix}10 & 20 & 30\\ 2 & 3 & 5\\ 3 & 5 & 0\end{pmatrix}=80 $
- If you transpose a matrix (ie turn the rows into columns, and columns into rows), the determinant stays the same.
- If you multiply matrices and take the determinant, the result is the same as taking the determinants, and then multiplying the determinants. $ \text{det}(A)*\text{det}(B)=\text{det}(A*B) $.
- It follows that det(A^{-1})=1/det(A).
- It also follows from the above statement that if a matrix $ B=S^{-1}AS $, then
$ \begin{align}\text{det}(B) & =\text{det}(S^{-1}AS) \\ & =\text{det}(S^{-1})*\text{det}(A)*\text{det}(S) \\ & =\text{det}(S)/\text{det}(S)*\text{det}(A) \\ & =\text{det}(A). \\ \end{align} $
