📘 Short Note: Matrices and Determinants

🔹 Matrix:
A matrix is a rectangular array of numbers arranged in rows and columns.
Example:
A = [ [1, 2],
    [3, 4] ]
(This is a 2×2 matrix)

🔹 Types of Matrices:

• Row Matrix – Only 1 row

• Column Matrix – Only 1 column

• Square Matrix – Rows = Columns

• Diagonal Matrix – Only diagonal elements are non-zero

• Identity Matrix (I) – Diagonal elements are 1, others are 0

🔹 Matrix Operations:

1. Addition/Subtraction – Same size only

2. Multiplication – Rows of 1st × Columns of 2nd

3. Transpose (Aᵗ) – Rows become columns

4. Inverse (A⁻¹) – A⁻¹ = adj(A) / |A| (Only for square matrices with |A| ≠ 0)

🔹 Determinant (|A|):
For 2×2 matrix:
If A = [ [a, b],
    [ c, d] ]
Then |A| = (a × d) − (b × c)

If |A| = 0 → No inverse (Singular matrix)

🔹 Applications:
✅ Solving linear equations
✅ Engineering systems (Electrical, Civil, Mechanical)
✅ Graphics and simulations
✅ Machine learning and data science