Matrices

What matrices covers

Grids of numbers with their own arithmetic, and a second life as movements of space. Transformations compose by multiplication, undo by inverses, and reveal their scaling and their fixed lines under inspection. Core Pure content, examinable on both compulsory papers of 9FM0, and the foundation for the eigenvalue and diagonalisation work in Further Pure 2.

The main ideas

• Addition and multiplication with the size rules, non-commutativity, and the zero and identity matrices.
• A matrix as a linear transformation, with its columns the images of the unit vectors.
• The plane catalogue: rotations about the origin, reflections in the axes and in y = x and y = -x, stretches parallel to an axis, and enlargements.
• Composition as multiplication with the first-applied matrix nearest the vector, and the standard reflections and rotations in three dimensions.
• Determinants of 2 by 2 and 3 by 3 matrices as area and volume scale factors, singular matrices, and inverses including the 3 by 3 case.
• Simultaneous equations written as a matrix equation and solved by the inverse, with the three-plane geometry when the determinant is zero.
• Invariant points and invariant lines of a transformation, and the difference between the two.

The results it turns on

det = ad − bc for a 2 × 2 matrix

the area scale factor, with the sign recording orientation

the inverse of rows (a, b), (c, d) is (1/det) times rows (d, −b), (−c, a)

the 2 by 2 inverse, available when the determinant is non-zero

minors, signs, transpose, then divide by the determinant

the 3 by 3 inverse, where the transpose is the step most often missed

Mv = b solved as v = M⁻¹b

a system of linear equations in matrix form

rotation anticlockwise by θ has columns (cos θ, sin θ) and (−sin θ, cos θ)

the booklet rotation, with a clockwise question using -theta

(AB)⁻¹ = B⁻¹A⁻¹

the reversal rule for the inverse of a product

Where it usually goes wrong

• Matrix multiplication does not commute in general, and doing B then A is the product AB, with the first-applied matrix written on the right.
• A product of two matrices can be the zero matrix without either factor being zero, so a matrix may be cancelled off both sides of an equation only once it is known to be invertible.
• A zero determinant flattens the plane onto a line or a point, which is why there is no inverse. For three planes it also separates the sheaf from the prism, and only the right-hand sides tell the two apart.
• An invariant line survives as a whole set, while a line of invariant points survives point by point. Say which of the two you have shown.

Where to start

Matrix algebra and transformations first, since everything else needs fluent multiplication. Determinants and inverses next, and verify one inverse by multiplying back to the identity. Systems and invariance last. The matrix induction proof in Further proof fits naturally after the first lesson.