• 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}

## Alumni Liaison

To all math majors: "Mathematics is a wonderfully rich subject."

Dr. Paul Garrett