The [[Determinant]] is a [[Function]] that takes in a Square [[Matrix]] and results in a [[Scalar]]. $\huge \op{\det} : M_{n\times n} \to \R $ The [[Determinant]] can for any dimensional space can be thought as how much does a [[Linear Transformation]] scale the [[Measure]] in $\R^n$ (ex. [[Area]], [[Volume]]). The most common use of the [[Determinant]] is the determine the [[Inverse Matrices|Invertibility]] of a given [[Matrix]]. If the [[Determinant]] of a [[Matrix]] is negative, then the corresponding transformation reverses [[Chirality]]. ## Properties [[Universal Quantifier|For Any]] two matrices $A,B \in M_{n\times n}$ and any [[Real Numbers|Real Number]] $k$: The [[Determinant]] of the [[Matrix Product|Product of Matrices]] is the product of their [[Determinant|Determinants]]. $\huge \det AB = \det (A) \det (B) $ The [[Determinant]] of the [[Identity Matrix]] is $1$ $\huge \det I_{n} = 1$ The [[Determinant]] of a [[Matrix Transpose]] is the [[Determinant]] of the [[Matrix]] $\huge \det A^\intercal = \det A $ The [[Determinant]] of a scaled [[Matrix]] is equal to the [[Determinant]] of the matrix scaled by $k^n$ $\huge \det{kA} = k^{n}\det{A} $ ## Methods of Computing #### $\R^2$ $\huge\begin{align*} \let A&= \mat{a&b\\c&d} \\ \det(A) &= \left| \begin{matrix} a&b\\c&d \end{matrix}\right| \\ &= ad - bc \end{align*}$ ### [[Augmented Matrix|Row Operations]] Applying a [[Augmented Matrix|Row Operation]] to some matrix $A$ leads to [[Elementary Matrix|easy to calculate]] results onto $As determinant. ### Cofactor Expansion A [[Recursion|Recursive]] [[Algorithm]] for computing the [[Determinant]] [[Universal Quantifier|for any]] $n\times n$ [[Matrix]] determinant using [[Minor Matrix|minor matrices]]. If $A\in M_{n\times n}$, then [[Universal Quantifier|for any]] row $i$ and column $j$, Then a cofacter $C_{i,j}$ of $A$ is: $ \huge C_{i, j} = (-1)^{i+j} \det\pa{M_{i,j}} \in \R $ Using such, the determinant of $A$ is: $\huge \det A = \partial_{1,1} C_{1,1} + \partial_{1,2}C_{1,2} + \cdots +\partial_{1,n}C_{1,n}$ $\huge \det A = \sum_{i=1}^{n} \partial_{1,i} C_{1, i} $ >[!example] $\Rn3$ >*$3\times 3$* Expansion >$ >\left|\begin{matrix} >\partial_{0,1} & \partial_{1,2} & \partial_{1,3}\\ >\partial_{2,1} & \partial_{2,2} & \partial_{2,3}\\ >\partial_{3,1} & \partial_{3,2} & \partial_{3,3}\\ >\end{matrix}\right| = >\partial_{1,1} \pa{\left|\begin{matrix} >\partial_{2,2} & \partial_{2,3} \\ >\partial_{3,2} & \partial_{3,3} \\ >\end{matrix}\right|} + >\partial_{1,2} \pa{-\left|\begin{matrix} >\partial_{2,1} & \partial_{1,3} \\ >\partial_{3,1} & \partial_{3,3} \\ >\end{matrix}\right|} + >\partial_{1,3} \pa{\left|\begin{matrix} >\partial_{2,1} & \partial_{2,2} \\ >\partial_{3,1} & \partial_{3,2} \\ >\end{matrix}\right|} >$ >[!tip]- Important Reminder $\huge (-1)^{i} = 👈👉👈👉👈👉👈👉 $