Matrix Operations — What Works (and what does'nt)

 

      📌Introduction

Not every operation between matrices is valid. In this post, we’ll cover:

🔹 How to add, subtract, and multiply matrices
🔹 When these operations are allowed
🔹 How to multiply a matrix by a scalar
🔹 Step-by-step examples

📌1. Matrix Addition

You can only add matrices of the same order. Addition is done element by element. If A = [aᵢⱼ] and B = [bᵢⱼ] have the same order, the sum is:

                                                C = A + B = [aᵢⱼ + bᵢⱼ]

Example:     

                                                   

📌2. Matrix Subtraction

Subtraction also applies only to matrices of the same order.

                                               C = A − B = [aᵢⱼ − bᵢⱼ]

Example:

📌3. Scalar Multiplication

Each element of the matrix is multiplied by a real number k. Given scalar k and matrix A, the product is kA.

Example:

   📌4. Matrix Multiplication

You can only multiply A × B if the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then the product C = AB will be of order m × p.

Example:

📌5. Warning: Matrix Multiplication is Not Commutative

Even when AB and BA both exist, in general AB ≠ BA.

    📌Conclusion 

Working with matrices is easier than applying formulas — as long as you know when the operation is valid and how to perform it properly.

📘 In the next post: Matrix Inversion — definition, how to compute (2×2), and solving AX = B.