Matrices are a way of organising data into a grid of rows and columns. They have their own set of rules for addition, subtraction, and multiplication.

1. Key Concepts & Vocabulary

• Matrix: A rectangular array of numbers arranged in rows and columns, enclosed in brackets.

• Element: Each individual number within a matrix.

• Order of a Matrix: Describes the size of the matrix. It is always written as (number of rows) × (number of columns).

  • A matrix with 3 rows and 2 columns is of order 3 × 2.

Types of Matrices

• Row Matrix: A matrix with only one row (e.g., [5 8 3]). Order is 1 × n.

• Column Matrix: A matrix with only one column. Order is m × 1.

• Square Matrix: A matrix where the number of rows equals the number of columns (e.g., 2×2, 3×3).

• Identity Matrix (I): A special square matrix with 1s on the main diagonal (top-left to bottom-right) and 0s everywhere else. It behaves like the number 1 in multiplication.

  • I of order 2x2 is [1 0; 0 1]

• Symmetric Matrix: A square matrix where the elements are symmetrical across the main diagonal.

2. Core Operations with Matrices

Addition and Subtraction

• The Golden Rule: You can only add or subtract matrices if they have the exact same order.

• Method: Simply add or subtract the elements in the corresponding positions.[a b; c d] + [e f; g h] = [a+e b+f; c+g d+h]

Scalar Multiplication (Multiplying a Matrix by a Number)

• Method: Multiply every single element inside the matrix by the number outside.k * [a b; c d] = [ka kb; kc kd]

Matrix Multiplication (The Tricky One!)

This is not like normal multiplication. You multiply row by column.

• The Rule for Multiplication (A × B): The number of columns in the first matrix (A) must be equal to the number of rows in the second matrix (B).

• How to Check: Write the orders side-by-side. If the two inner numbers match, you can multiply.

  • A (m × n) × B (n × p) -> The n's match, so it's possible.

• Order of the Result: The resulting matrix will have the order of the two outer numbers.

  • In the example above, the result will be m × p.

• Method: "Row into Column" To find the element in Row 1, Column 1 of the answer, you multiply the elements of Row 1 (first matrix) by the elements of Column 1 (second matrix) and add them up. Repeat for every position.

Exam Tips & Common Pitfalls

• Pitfall 1: Matrix Multiplication Order Matters! In general, A × B is NOT the same as B × A. This is the most common mistake. Always calculate in the exact order given in the question.

• Pitfall 2: Forgetting the Order Rule. Before multiplying, always check if it's possible by comparing the columns of the first matrix to the rows of the second. If they don't match, the product is "not defined".

• Tip 1: Solving for Unknowns. If you see an equation where two matrices are equal, it means every corresponding element is equal. You can create simple algebraic equations from this to find unknown values like x and y.

  • If [x 5; 1 y] = [3 5; 1 -2], then x = 3 and y = -2.

• Tip 2: Combining Operations. Questions often involve scalar multiplication, addition, and subtraction in one line (e.g., "Find 3A - 2B"). Do the scalar multiplication first to get the new matrices, then perform the subtraction.