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Step 4: the determinant of an $n\times n$ matrix
Definition
If $\def\mat#1{\begin{bmatrix}#1\end{bmatrix}}A=\mat{a_{11}&a_{12}&\dots&a_{1n}\\\vdots&&&\vdots\\a_{n1}&a_{n2}&\dots&a_{nn}}$ is an $n\times n$ matrix, then \[\det A=a_{11}C_{11}+a_{12}C_{12}+\dots+a_{1n}C_{1n}.\] Here $C_{ij}$ are the cofactors of $A$.
This formula is called the Laplace expansion of $\det A$ along the first row, since $a_{11}, a_{12},\dots,a_{1n}$ make up the first row of $A$.
Example
\begin{align*} \def\vm#1{\begin{vmatrix}#1\end{vmatrix}} \vm{1&0&2&3\\0&2&1&-1\\2&0&0&1\\3&0&4&2} &= 1\vm{2&1&-1\\0&0&1\\0&4&2}-0\vm{0&1&-1\\2&0&1\\3&4&2}+2\vm{0&2&-1\\2&0&1\\3&0&2}-3\vm{0&2&1\\2&0&0\\3&0&4}\\ &= 1\left(2\vm{0&1\\4&2}-1\vm{0&1\\0&2}-1\vm{0&0\\0&4}\right)-0+2\left(0-2\vm{2&1\\3&2}-1\vm{2&0\\3&0}\right)-3\left(0-2\vm{2&0\\3&4}+1\vm{2&0\\3&0}\right)\\ &=1(2(-4)-0-0)+2(-2(1)-0)-3(-2(8)+0)\\ &=-8-4+48\\ &=36. \end{align*}
Theorem
Let $A$ be an $n\times n$ matrix.
- $A$ is invertible if and only if $\det(A)\ne0$.
- If $A'$ is the same as $A$, except with two rows swapped, then $\det(A')=-\det(A)$.
- If $c$ is a scalar and $A'$ is the same as $A$ except with one row multiplied by $c$, then $\det(A')=c\det(A)$.
- If $A'$ and $A''$ are the same as $A$ except in row $i$, and $row_i(A'')=row_i(A)+row_i(A')$, then $\det(A'')=\det(A)+\det(A')$.
- $\det(A^T)=\det(A)$. So we can swap “row” with “column” in these properties.
- If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$.