Matrices — Study Notes

Defⁿ A matrix is an ordered rectangular array of numbers or functions. The numbers/functions inside are called elements or entries.

👉 Horizontal lines = rows | 👉 Vertical lines = columns

A = [ 2  5
   0  √5
   3  6 ]

→ 3 rows, 2 columns

Rule A matrix with m rows and n columns → order m × n (read "m by n").
Written as: A = [a_ij]_m×n, where a_ij = element in i^th row, j^th column.

Total elements in a matrix = m × n
To find possible orders → find factor pairs of total elements!
e.g. 8 elements → orders: 1×8, 8×1, 2×4, 4×2

🗂️ Quick-look table
Type	Meaning	Example
🔹 Row Matrix	Only 1 row → order 1×n	[ 5 2 3 ]
🔹 Column Matrix	Only 1 column → order m×1	[ 1 ; 4 ; 7 ]
🔹 Square Matrix	rows = columns (m = n)	3×3, 2×2 etc.
🔹 Diagonal Matrix	square matrix, all non-diagonal = 0	diag(4,2,3)
🔹 Scalar Matrix	diagonal matrix with all diagonal entries equal (=k)	diag(3,3,3)
🔹 Identity Matrix (I)	diagonal = 1, rest = 0	I₃
🔹 Zero / Null Matrix (O)	all elements = 0	[0 0;0 0]

📌 Note: Every identity matrix is a scalar matrix (k=1), but not every scalar matrix is an identity matrix!

Two matrices A = [a_ij] and B = [b_ij] are equal only if:

Same order (same rows & columns)
Every corresponding element is equal → a_ij = b_ij

Used to solve for unknowns (x, y, z...) by comparing corresponding entries! 🔍

1️⃣ Addition Add corresponding elements — only possible if matrices are of the same order.
A + B = [a_ij + b_ij]

2️⃣ Scalar Multiplication Multiply every element by the scalar k.
kA = [k·a_ij]
➡️ Negative of matrix: −A = (−1)A

3️⃣ Subtraction A − B = A + (−1)B (same order needed)

4️⃣ Multiplication Defined only if: columns of A = rows of B.
If A is m×n and B is n×p → AB is m×p
Each element: c_ik = Σ a_ij b_jk (row of A • column of B)

⚠️ Matrix multiplication is NOT commutative! Generally AB ≠ BA. Also, AB = O does not mean A = O or B = O! 😲

Addition

A + B = B + A (commutative)
(A+B)+C = A+(B+C) (associative)
A + O = O + A = A (identity)
A + (−A) = O (inverse)

Multiplication

(AB)C = A(BC) (associative)
A(B+C) = AB + AC (distributive)
(A+B)C = AC + BC (distributive)
IA = AI = A (mult. identity)

Defⁿ Formed by interchanging rows and columns of A.
If A = [a_ij]_m×n then A′ = [a_ji]_n×m

✨ Properties of Transpose
Property	Rule
Double transpose	(A′)′ = A
Scalar	(kA)′ = kA′
Addition	(A + B)′ = A′ + B′
Multiplication	(AB)′ = B′A′ (order flips! 🔁)

Symmetric A′ = A → a_ij = a_ji
(mirror image about diagonal!) 🪟

Skew-Symmetric A′ = −A → a_ji = −a_ij
💡 All diagonal elements = 0

🌟 Big Theorem: Any square matrix A can be written as the sum of a symmetric and a skew-symmetric matrix:

A = ½(A + A′) + ½(A − A′)
symmetric skew-symmetric

Square Matrix A

➡️

½(A+A′) → Symmetric

➕

½(A−A′) → Skew-Sym.

Defⁿ Square matrix A (order m) is invertible if ∃ square matrix B (same order) such that:
AB = BA = I
Then B is called A⁻¹ (inverse of A).

Only square matrices can have inverses
Inverse of a matrix (if it exists) is unique ✔️
(AB)⁻¹ = B⁻¹A⁻¹ (order reverses, just like transpose!)

✅ Quick Revision Checklist