Matrices · Level 2 Further Maths

Writing matrices

A matrix is a rectangular array of numbers. Its order is rows × columns: a 2 × 2 matrix has 2 rows and 2 columns; a column vector is 2 × 1.
In these notes, [2 1; 3 4] means the matrix with top row 2 1 and bottom row 3 4. [5; 6] is a column vector with 5 on top and 6 below.

Multiply each row of the first matrix by each column of the second: multiply matching entries and add.
[2 1; 3 4] × [5; 6] = [2×5 + 1×6; 3×5 + 4×6] = [16; 39].
[1 2; 3 4] × [0 1; 1 0] = [1×0 + 2×1, 1×1 + 2×0; 3×0 + 4×1, 3×1 + 4×0] = [2 1; 4 3].
You can only multiply when the number of columns in the first equals the number of rows in the second. A 2 × 2 times a 2 × 1 gives a 2 × 1; a 2 × 1 times a 2 × 2 isn't possible.
Order matters: in general, AB ≠ BA. A² means A × A.

The identity matrix I = [1 0; 0 1] leaves any matrix unchanged: AI = IA = A.
To multiply by a number (a scalar), multiply every entry: 3 × [1 2; 0 −1] = [3 6; 0 −3].
To find an unknown entry, multiply out and compare entries with the answer.

Matrix: A rectangular array of numbers.
Order: The size of a matrix, given as rows × columns.
Column vector: A matrix with one column.
Identity matrix: The matrix [1 0; 0 1], which leaves other matrices unchanged.
Scalar: A single number that multiplies every entry of a matrix.
Entry: One number in a matrix.