Matrices and Transformations · Mathematics

Matrices

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns is an m × n matrix. Two matrices of the same size are added or subtracted element by element. To multiply by a scalar, multiply every element.

Multiply rows by columns. The number of columns in the first must equal the number of rows in the second.

Worked exampleMultiplying matrices

Find the product of (2 1; 0 3) and (1 4; 2 5).

Row 1 × columns.
Row 2 × columns.

Answer: The product is (4 13; 6 15). Note that AB is usually not equal to BA.

Determinant and inverse

A matrix has an inverse only if its determinant is not zero.

Matrices as transformations

A 2 × 2 matrix transforms points of the plane. The point (x, y) maps to (ax + by, cx + dy). Common matrices are: reflection in the x-axis (1 0; 0 −1), reflection in the y-axis (−1 0; 0 1), reflection in y = x (0 1; 1 0), rotation through 90° anticlockwise about the origin (0 −1; 1 0), and enlargement of scale factor k (k 0; 0 k). The determinant is the area scale factor.

Try itMatrix transformation of a point

Choose a matrix and a point. See the image and the determinant.

Check yourself The determinant of (3 2; 1 4) is

Check yourself The matrix (1 0; 0 −1) is a reflection in the

Check yourself A matrix with determinant 0...

Key terms

Matrix: A rectangular array of numbers.
Determinant: ad − bc for a 2 × 2 matrix.
Inverse matrix: A matrix that undoes another.
Identity matrix: The matrix (1 0; 0 1) that changes nothing.

Summary

Add matrices of equal size; multiply row by column.
det = ad − bc; the inverse exists only if det ≠ 0.
Matrices represent reflections, rotations and enlargements.
The determinant is the area scale factor.