🗂️Introduction to Matrices

Data lined up in rows and columns forms a matrix — a tidy table of numbers. The pixels on your screen, a league table, the score sheets of three classes: all relatives of matrices. Do not underestimate the table — it can do arithmetic on its own, and it can move an entire figure in one sweep.

Matrix addition and scalar multiplication

Adding two matrices of the same shape follows a rule too plain to memorize: add the entries at matching positions.

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}$

Scalar multiplication also goes cell by cell: $2 \times \begin{pmatrix} 1 & 0 \\ 3 & 2 \end{pmatrix} = \begin{pmatrix} 2 & 0 \\ 6 & 4 \end{pmatrix}$. Matrices of different shapes cannot be added — the positions do not even line up.

Row times column: the main event

The coefficients of simultaneous equations arrange into a matrix as old partners; multiplication itself, though, refuses to work cell by cell. Its rule is row times column: the entry in row i, column j of the result comes from row i of the left matrix meeting column j of the right one — multiply pairwise, then add. Watch:

$\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 1\cdot1 + 2\cdot2 & 1\cdot1 + 2\cdot0 \\ 0\cdot1 + 1\cdot2 & 0\cdot1 + 1\cdot0 \end{pmatrix} = \begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix}$

The top-left 5 comes from row $(1, 2)$ meeting column $(1, 2)$: $1\times1 + 2\times2 = 5$. The other three cells follow the same recipe.

Order matters

Among plain numbers $3 \times 4 = 4 \times 3$; matrices lose that freedom. Swap the order of the two matrices above:

$\begin{pmatrix} 1 & 1 \\ 2 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$

Earlier the answer was $\begin{pmatrix} 5 & 1 \\ 2 & 0 \end{pmatrix}$; now it comes out $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ — swap the order and the whole answer changes. In the matrix world, $AB = BA$ generally fails.

Moving shapes with matrices

The strange row-times-column rule reveals its true meaning in geometry. In the coordinate plane, write the point $(x, y)$ as a column and let a matrix multiply it from the left:

$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -y \\ x \end{pmatrix}$

The new point is $(-y, x)$ — exactly a 90° counterclockwise rotation about the origin. Try two points: $(1, 0)$, pointing due east, turns to $(0, 1)$, due north; $(2, 3)$ turns to $(-3, 2)$.

A matrix plus the instruction "multiply by me" is a geometric command. Beyond rotation come reflections and stretches — the whole repertoire of Transformations is within a matrix's reach.

Check yourself

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Quick quiz

↺ Try again

1. 1. What is [1,2,3,4] plus [5,6,7,8] (two rows, two columns, listed by rows)?

   [6,8,10,12][19,22,43,50][6,8,10,10]

2. 2. Does matrix multiplication satisfy AB = BA?

   always, just like ordinary numbersgenerally not — order is crucialonly when the matrices are big

3. 3. Using the rotation matrix, turn the point (2, 1) counterclockwise by 90°. What do you get?

   (-1, 2)(1, -2)(1, 2)