📘 Chapter 7: Determinants

📐 What is a Determinant?

The determinant of a square matrix \( A \) is a scalar value that can be computed from its elements and captures key properties such as invertibility, volume scaling, and orientation.

🔎 Properties

• \( \det(A) \ne 0 \) ⇒ Matrix is invertible
• \( \det(A^T) = \det(A) \)
• \( \det(AB) = \det(A) \cdot \det(B) \)
• Row swaps flip the sign of the determinant

⚙️ Methods to Compute

• 2×2 Matrix: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad \det(A) = ad - bc \]
• 3×3 or Higher: Use cofactor expansion or row reduction

Laplace Expansion (Cofactor Expansion)

Pick any row/column and compute: \[ \det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij} \] where \( M_{ij} \) is the minor of \( a_{ij} \)

📊 Example

Compute the determinant of \[ A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \Rightarrow \det(A) = 2 \cdot 4 - 3 \cdot 1 = 5 \]

📝 Practice

Q: Find the determinant of \[ B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix} \]