Determinant - Yousef's Notes
Determinant

Determinant

The determinant of a transformation is a scalar value that can be used to describe the scaling effect of the transformation on a region of space. Given a $2 \times 2$ matrix: $$ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$ The determinant of $A$ is denoted as $\det(A)$ or $|A|$ and is calculated as: $$ \det(A) = ad - bc $$
  • Matrix is singular when $det(A) = 0$
  • Notation: $det(A) = \abs{A}$
  • Matrix is not singular or invertible when $det(A) \neq 0$

#Determinant Diagonal Method

Only works for matrices lower than 4x4.

#Determinant Properties

  1. Det I = 1
  2. Exchange Rows reverse sign: det P = 1 even or -1 odd
  3. a)
$$ \begin{vmatrix} ta & tb \\ c & d \end{vmatrix} = t \begin{vmatrix} a & b \\ c & d \end{vmatrix} $$ 
b)
$$ \begin{vmatrix} a + a' & b + b' \\ c & d \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} a' & b' \\ c & d \end{vmatrix} $$ 
3a and 3. b means linear each row
  1. If 2 rows are equal then det = 0 Exchange those rows equal so you have the same matrix. Determinant should change sign by property 2.
  2. Subtract m times row i from row k then determinant doesn’t change:
$$ \begin{vmatrix} a & b \\ c - ma & d - mb \end{vmatrix} = \begin{vmatrix} a & b \\ c & d \end{vmatrix} + \begin{vmatrix} a & b \\ -ma & -mb \end{vmatrix} = (by\ 3.b) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} $$
  1. Rows of zeros then Det A = 0
$$ \begin{vmatrix} 0 & 0 \\ c & d \end{vmatrix} = (by\ 3.a) = \begin{vmatrix} 0 \cdot a & 0 \cdot b \\ c & d \end{vmatrix} = \begin{vmatrix} 0 & 0 \\ c & d \end{vmatrix} $$
  1. Determinant of Upper Triangular Matrix then determinant is the product of the elements of diagonal
$$ U = \begin{bmatrix} d_1 & * & * & * & * \\ 0 & d_2 & * & * & * \\ 0 & 0 & .. & * & * \\ 0 & .. & 0 & .. & * \\ 0 & .. & .. & 0 & d_n \end{bmatrix} \quad \text{det } U = \begin{bmatrix} d_1 & * & * & * & * \\ 0 & d_2 & * & * & * \\ 0 & 0 & .. & * & * \\ 0 & 0 & 0 & .. & * \\ 0 & .. & .. & 0 & d_n \end{bmatrix} = (d_1) \cdot (d_2) \cdot \cdots (d_n) $$
  1. $\text{det } A = 0$ when $A$ is singular $\rightarrow$ row of zeros $\text{det } A \neq 0$ when $A$ is invertible
  2. Det (AB) = (Det A)(Det B)
  3. $\text{det } A^T = \text{det } A$

#Different Ways to Calculate Determinants

Let $A = \begin{bmatrix} 2 & 4 & 4 \ 1 & 2 & 1 \ 3 & -2 & 5 \end{bmatrix}$ then solve $\det A = \begin{vmatrix} 2 & 4 & 4 \ 1 & 2 & 1 \ 3 & -2 & 5 \end{vmatrix}$

  • Using Diagonal formula
  • Using Cofactors of Row 1

    • $\text{Det } A = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$

  • Using Cofactors of Column 3
    • $\text{Det } A = a_{13}C_{13} + a_{23}C_{23} + a_{33}C_{33}$

#Applications

Hill Cipher