Determinant

Definition

$\text{det}(\mathbf{A})=|\mathbf{A}|$

Computation

$2\times 2$ matrix

$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

$n\times n$ matrix

Laplace Expansion

Definition

Let $C$ be a cofactor matrix of $\mathbf{A}$

Along the $j$th column gives

$\text{det}(\mathbf{A})={\sum }_{i=1}^n{a}_{ij}{C}_{ij}={\sum }_{i=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{i,j}$

Along the $i$th row gives

$\text{det}(\mathbf{A})={\sum }_{j=1}^n{a}_{ij}{C}_{ij}={\sum }_{j=1}^n{a}_{ij}(-1{)}^{i+j}{M}_{i,j}$

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Facts

$\text{det}\mathbf{A}\ne 0\iff \exists {\mathbf{A}}^{-1}$ $\text{det}\mathbf{A}=0\iff \nexists {\mathbf{A}}^{-1}$

The volume of box^[parallelepiped] $\in {\mathbb{R}}^n$ is expressed by a determinant

$\text{det}(\mathbf{I})=1$

$\text{det}(\mathbf{A})$ changes sign, when two rows are exchanged (permutated)

$\text{det}(\mathbf{A})$ depends on linearly on the 1st row.

$\text{det}(t\mathbf{A})=t^n\text{det}(\mathbf{A})$

$\text{det}(\mathbf{AB})=\text{det}(\mathbf{A})\text{det}(\mathbf{B})$

If two row or column vectors in $\mathbf{A}$ are equal, then $\text{det}(\mathbf{A})=0$

Gauss Elimination does not change $\text{det}(\mathbf{A})$

If a matrix $\mathbf{A}$ has zero row or column vectors, then $\text{det}(\mathbf{A})=0$

If a matrix $\mathbf{A}$ is diagonal or triangular, then the determinant of $\mathbf{A}$ is the product of diagonal elements $\text{det}(\mathbf{A})={\prod }_{i=1}^n{a}_{ii}$ where $a_{ii}$ is the $i,i$ element of the matrix $\mathbf{A}$.

$\text{det}({\mathbf{A}}^{-1})=\text{det}(\mathbf{A}{)}^{-1}$

$\text{det}({\mathbf{A}}^{⊺})=\text{det}(\mathbf{A})$

$|\mathbf{A}|={\prod }_{i=1}^n{\lambda }_i$ The Determinant of the matrix is equal to the product of the eigenvalues of the matrix.

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$|I_m+AB|=|I_n+BA|$ where the dimensions of the matrices are $m\times n$ and $n\times m$ respectively.

$|A\pm \mathbf{u}{\mathbf{u}}^{⊺}|=|A|(1\pm {\mathbf{u}}^{⊺}{A}^{-1}\mathbf{u})$