Linear Algebra
Class 12

Matrix Multiplication

[AB]ij=kaikbkj[AB]_{ij} = \sum_k a_{ik} b_{kj}
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[AB]ᵢⱼ = Σₖ aᵢₖ × bₖⱼ

Row of first matrix × column of second matrix. A(m×n) × B(n×p) = C(m×p). Not commutative: AB ≠ BA generally.

Real-World Applications

Computer graphics — Chaining transformations (rotate, scale, translate).

Machine learning — Neural network forward pass: output = W × input.

Economics — Input-output analysis: X = (I-A)⁻¹D.

Markov chains — State transitions: P^n gives n-step probabilities.

Cryptography — Hill cipher encryption.

Quantum computing — Quantum gate operations.

Signal processing — Filter banks and convolutions.

Network analysis — Path counting in graphs.

Robotics — Forward kinematics: joint angles to end effector.

Google PageRank — Web page importance via matrix powers.

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