Transpose and Determinant

Transpose

Determinant

Determinant

The determinant of a square matrix $A$ is $\det A:={\sum }_{\sigma \in S_n}\mathrm{sgn}(\sigma ){\prod }_{k=1}^n{A}_{k,\sigma (k)},$ where $S_n$ is symmetric group, $\sigma$ is a permutation, $\text{sgn}(\sigma )$ is the sign of the permutation, and $A_{k,\sigma (k)}$ denotes the $(k,\sigma (k))$ entry of $A$.

Cofactor Expansion

For any $i$, $\det A={\sum }_{j=1}^n{A}_{ij}{C}_{ij}$, where $C_{ij}=(-1{)}^{i+j}\det (A_{\hat{i}\hat{j}})$, and $A_{\hat{i}\hat{j}}$ denotes the matrix obtained from $A$ by removing the $i$th row and $j$th column.

e.g. For a $3\times 3$ matrix $A=\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$, the determinant can be recursively computed as follows: $\begin{array}{rl}\det A& =a\det \left(\begin{array}{cc}e& f\\ h& i\end{array}\right)-b\det \left(\begin{array}{cc}d& f\\ g& i\end{array}\right)+c\det \left(\begin{array}{cc}d& e\\ g& h\end{array}\right)\\ & =a(ei-fh)-b(di-fg)+c(dh-eg)\end{array}$

Proposition

Determinant satisfies the following properties:

• $\det I=1$.
• If $B$ is obtained from $A$ by scaling a single row by a factor $\alpha \in \mathbb{R}$, then $\det B=\alpha \cdot \det A$. In particular, $\det \alpha A={\alpha }^n\det A$.
• If all rows of $A$, $B$, $C$ except the $i$th are same, and for any $j$, $C_{ij}=A_{ij}+B_{ij}$, then $\det C=\det A+\det B$.
• If $B$ is obtained by swapping 2 rows of $A$, then $\det B=-\det A$.
• If $B$ is obtained by adding $\alpha \cdot A_i$ to $A_j$ for $i\ne j$, then $\det B=\det A$.
• $\det AB=\det BA=\det A\det B$.