FP3 · Chapter 6
Matrices
A matrix isn't just a grid of numbers — it's a machine that transforms space.
Once you can find its determinant, invert it, and see what it does to lines and planes,
you can predict exactly how any shape moves, stretches or flattens under that transformation.
Summary
- Dimension of a matrix = (rows) × (columns).
- Transpose AT swaps rows and columns; a symmetric matrix satisfies S = ST.
- Identity I leaves everything unchanged; the zero matrix sends everything to zero.
- The determinant of a 3×3 matrix is found by cofactor expansion along any row or column.
- A matrix is singular if det = 0 (no inverse exists), non-singular if det ≠ 0.
- The inverse of a 3×3 matrix is built from the matrix of cofactors, transposed, then divided by the determinant.
- Matrices represent linear transformations: functions that preserve addition and scalar multiplication.
- The columns of a transformation matrix are exactly where the base vectors i, j, k land.
- To find the image of a line, apply the transformation to the position vector and the direction vector separately.
- To find the image of a plane, first rewrite it in vector form r = a + λb + μc, then transform each piece.
1. Basic Definitions
Dimension of a matrix
A matrix with r rows and c columns has dimension r × c
(rows always come first — "Railway Carriage" is the classic way to remember the order).
So a matrix with 3 rows and 2 columns is a "3 × 2" matrix. In FP3 you'll mostly work with
3 × 3 square matrices, since those represent transformations of 3-D space.
Transpose and symmetric matrices
The transpose of a matrix A, written AT, is found by turning rows into
columns (or equivalently, reflecting every entry in the leading diagonal).
A really important (and easy to forget) rule is what happens when you transpose a product:
Transpose of a product
(AB)T = BTAT — the order flips.
A matrix S is symmetric if it looks the same when reflected in its leading
diagonal — formally, if S = ST. Visually, the entry in row i, column j
always matches the entry in row j, column i.
Example — spotting a symmetric matrix
S =
is symmetric — notice the off-diagonal entries mirror each other perfectly:
top-right = bottom-left in each pair.
Identity and zero matrices
The identity matrix I is the "do nothing" transformation — multiplying any
vector or matrix by I leaves it unchanged, just like multiplying a number by 1.
The zero matrix sends every vector to the origin, just like multiplying by 0.
Practice — Q1
Write down the transpose of B =
[[2,-1,4],[0,3,5],[7,1,-2]] (rows given left to right, top to bottom),
and state whether B is symmetric.
2. Determinant of a 3 × 3 Matrix
The determinant is a single number attached to a square matrix that tells you how the
transformation scales volume (and whether it flips orientation, if it's negative). You'll
already have met 2 × 2 determinants — the 3 × 3 version builds directly on that by
expanding along a row using smaller 2 × 2 determinants.