Determinant

Meaning

• Describes the impact of the transformation represented by the matrix
• Magnitude: How much it “scales” what a shape would enclose in the dimension
• Sign: if the handedness of the coordinate system flips - Right handed: if $u$ can be rotated counterclockwise by $\theta \in (0,\pi )$ to coincide with $v$. Left handed otherwise

Notation

$$\det \left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|$$

Finding

2x2

$$\det \left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)=a_{11}{a}_{22}-a_{21}{a}_{12}$$

3x3

$$\begin{array}{rl}& \det \left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)\\ & =a_{11}\det \left(\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right)-a_{12}\det \left(\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right)+a_{13}\det \left(\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right)\end{array}$$

nxn

• Let ${\tilde{A}}_{ij}$ be A with row $i$ and column $j$ deleted
• $\det (A)={\sum }_{j=1}^n(-1{)}^{i+j}{A}_{ij}\det ({\tilde{A}}_{ij})$ is the cofactor expansion along row $i$
  • $(-1{)}^{i+j}\det ({\tilde{A}}_{ij})$ is the cofactor

Shortcuts

• If $A$ has a row of zeros, then $\det (A)=0$
  • → if $\text{rank}(A)<n⟹\det (A)=0$
• If row $i$ of $B=e_k$ for some $k\in (1,n)$, then $\det (B)=(-1{)}^{i+k}\det ({\tilde{B}}_{ik})$
• If $A$ upper triangular, then $\det (A)={\prod }_{i=1}^n{A}_{i,i}$ (product of diagonal elements)

Properties

nxn

• $\det$ is a linear function of each row when the remaining rows are held fixed
• Impact of Elementary Matrix Operations
  1. Swapping rows: $\det (B)=-\det (A)$
  2. Multiplying a row by $k\ne 0$: $\det (B)=k\det (A)$
  3. Adding a multiple: $\det (B)=\det (A)$
• $\det (AB)=\det (A)\det (B)$
• $A$ invertible $⟺\det (A)\ne 0$
  • $A$ invertible $⟹\det (A^{-1})=\det (A{)}^{-1}$
• $\det (A^t)=\det (A)$

2x2

• Area of parallelogram determined by $u,v$ is $A\left(\begin{array}{c}u\\ v\end{array}\right)=\left|\det \left(\begin{array}{c}u\\ v\end{array}\right)\right|$
• $\det (A)\ne 0⟺A$ invertible
  • Moreover, $A^{-1}=\frac{1}{\det (A)}\left(\begin{array}{cc}{A}_{22}& -A_{12}\\ -A_{21}& A_{11}\end{array}\right)$