Matrix Operations

Matrix Addition Rules

(A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ

Same dimensions required: You can only add or subtract matrices that have the exact same number of rows and columns. A 2×3 matrix cannot be added to a 3×2 matrix.

Element-wise operation: Each entry in the result comes from adding (or subtracting) the corresponding entries. Row 1, Column 1 of A pairs with Row 1, Column 1 of B.

Commutative: A + B = B + A. Order does not matter for addition.

Associative: (A + B) + C = A + (B + C). You can group additions freely.

Scalar multiplication works for any matrix of any size — you just multiply every single entry by the scalar c.

When Operations Fail

(A+B) undefined if dims differ

A + B undefined: If A is m×n and B is p×q, addition is only defined when m = p and n = q. Dimension mismatches cause an error — not just a "bad answer."

Scalar multiply: c · A always works. Every element gets scaled: (cA)ᵢⱼ = c · Aᵢⱼ. Note that c = −1 negates the matrix, turning every entry into its opposite.

Transpose always works: Any m × n matrix can be transposed to give an n × m matrix. Rows become columns: (Aᵀ)ᵢⱼ = Aⱼᵢ.

A − B = A + (−1) · B — subtraction is addition of the negation.
If A = Aᵀ, the matrix is symmetric (common in real-world data).
If Aᵀ = −A, the matrix is skew-symmetric; its diagonal must be zero.
Transposing twice returns the original: (Aᵀ)ᵀ = A.

Matrix operations still confusing?