Summary of Determinants

🎨 Determinants Overview

• Determinants provide a value for square matrices (only applicable for square matrices). 🎲

• The determinant is used in linear algebra to determine properties of matrices, such as invertibility. 🔑

🌟 Second Order Determinant

• For a 2x2 matrix:
  

  $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$

• The determinant is calculated as:
  

  $|A| = ad - bc$

• This corresponds to the difference between the product of the main diagonal and the product of the other diagonal. 🎉

✨ Example of 2x2 Determinant

• For matrix
  

  $\begin{pmatrix} 2 & 5 \ 3 & 8 \end{pmatrix}$:
  

  $|A| = 2 \times 8 - 3 \times 5 = 16 - 15 = 1$ 🎈

🔍 Finding Unknowns in Determinants

• Unknowns can appear in determinants; solve for the unknown via the determinant equation. 🔎

• Example:
  

  $x^2 - 4 \begin{pmatrix} 1 & 0 \ 1 & 0 \end{pmatrix} = 0$
  

  results in the quadratic equation:
  

  $x^2 - 4 = 0 \Rightarrow x = \pm 2$ 💡

🎈 Third Order Determinant

• For a 3x3 matrix:
  

  $A = \begin{pmatrix} a<em>{11} &amp; a</em>{12} &amp; a<em>{13} \ a</em>{21} &amp; a<em>{22} &amp; a</em>{23} \ a<em>{31} &amp; a</em>{32} &amp; a_{33} \end{pmatrix}$

• The determinant is denoted as:
  

  $|A|$

• To compute, understand the concept of minor determinants associated with elements of the matrix. 🌟

🧩 Minor Determinants

• Minor is obtained by deleting the row and column of a given element.

• Sign of the minor determinant depends on the order of the row and column:

  • Even: Positive (+) ✨

  • Odd: Negative (−) ⚠

✔ Sign Rule for Minors

• Example: - Element: $a_{11}$ → sign is + (1 + 1 = 2, even)

  • Element: $a_{12}$ → sign is − (1 + 2 = 3, odd)

🎉 Expanding the Third Order Determinant

• Expanding can use any row or column: - $|A| = a<em>{11} \cdot M</em>{11} - a<em>{12} \cdot M</em>{12} + a<em>{13} \cdot M</em>{13}$

  • Where $M_{ij}$ are minor determinants. 🍀

🌈 Determinant of Triangular Matrices

• Value equals the product of the diagonal elements. 💫

🚀 Application: Area of Triangle Using Determinants

• Determine area using vertices coordinates: - If vertices are $X(a, b), Y(c, d), Z(e, f)$ then:
  

  $Area = \frac{1}{2} |A|$

  • Area calculated as the absolute value. 📐