MST10030 notes wiki

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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 $A$.

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 $\det(AB)=\det(A)\det(B)$.