Determinant

A matrix operation that is useful for determining scale factor that a transformation has on the area of the ‘paralellogram’ created by a linear transformation on its basis vectors.

Definition

The determinant is the alternating Multilinear Map function that sends the identity matrix to 1.

Formal Definition

1. Consider $M_{n\times n}$ as $({\mathbb{R}}^n{)}^m$ where the rows are thought as vectors in ${\mathbb{R}}^n$
2. If $f:M_{n\times n}(\mathbb{R})\to \mathbb{R}$ is
   1. Alternating
   2. Multilinear Map
   3. Satisfies $f(I)=1$ Then, $f=\det$

Multilinear Properties

Normalization Property

$\det (I)=1$ Where $I$ is the Identity Transformation

Product Property

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

Exponential Property

$\det (A^k)=(\det (A){)}^k$

Constant Multiple Property

If $A$ is $n\times n$ matrix $\det (kA)=k^n\det (A)$

Transpose Property

$\det (A)=\det (A^T)$ where $A^T$ is the matrix transpose of $A$.

Invertability Property

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

Interchanging Property

Interchanging two rows or columns multiplies the determinant by $(-1)$

Row Multiplication Property

Multiplying a row by $k$ multiplies the determinant by $k$

Linear Combination Property

Adding a multiple of one row to another does not change the value of the determinant

Identical Rows Property

If two rows are identical than the determinant is zero

Theorems

• Determinants and Invertiblity Theorem
• Determinants and Row Operations
• Recursive Formula for Determinant
• Determinants and Eigenvalues
• Determinant from Permutations

Concepts

• Diagonal Matrices
• Determinant of 1x1 Matrix
• Determinant of 2x2 Matrix
• Determinant of Diagonal Matrix