Determinants of Matrices of Different Order

Order 1. For $A=[a]$, $|A|=a$ — simply the single entry.

Order 2. For $A=\begin{pmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{pmatrix}$, $|A|=a_{11}a_{22}-a_{12}a_{21}$ — product of the main-diagonal pair minus product of the other diagonal pair. E.g. $\begin{vmatrix}2&4\\-1&2\end{vmatrix}=(2)(2)-(-1)(4)=4+4=8$, and $\begin{vmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{vmatrix}=\cos^2\theta+\sin^2\theta=1$.

Order 3 — minors and cofactors. For $A=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$, the minor $M_{ij}$ of entry $a_{ij}$ is the order-2 determinant obtained by deleting row $i$ and column $j$; the cofactor is the signed minor $A_{ij}=(-1)^{i+j}M_{ij}$. For instance, $M_{11}=\begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}=a_{22}a_{33}-a_{32}a_{23}$ and $A_{11}=(-1)^{1+1}M_{11}=M_{11}$, while $A_{12}=(-1)^{1+2}M_{12}=-M_{12}$.

Laplace expansion (Result 7.1/7.2). $|A|$ equals the sum of the products of the entries of any one row (or column) with their corresponding cofactors, and this value is the same no matter which row/column you choose:

and equally $|A|=a_{21}A_{21}+a_{22}A_{22}+a_{23}A_{23}$ (row 2), or $|A|=a_{11}A_{11}+a_{21}A_{21}+a_{31}A_{31}$ (column 1), etc. For hand computation, expand along whichever row/column has the most zeros.

Sarrus' Rule (a direct order-3 shortcut, named after Pierre Frédéric Sarrus): repeat the first two columns to the right, then

(the three products running down-right minus the three products running down-left).

Worked illustration. For $A=\begin{pmatrix}1&3&-2\\4&-5&6\\3&5&-2\end{pmatrix}$ (numbers per the textbook's own example), expanding along row 1 with the cofactors $A_{11}=-40, A_{12}=-26, A_{13}=5$ gives $|A|=1(-40)+3(-26)+(-2)(5)=-40-78-10=-128$; expanding instead along column 1 gives the same value $-128$, confirming Laplace expansion is expansion-path independent. …