Speak Maths Β· University
Matrices: a recipe for many cakes
A matrix is a table of numbers that holds a whole recipe. A few signs say how to read it, undo it, and spot what it only stretches.
Draft: written by an AI helper and not yet checked by a teacher.
The alphabet
Read each sign aloud, the way it is written under it.
- Say: the matrix A, with entries a one one, a one two, a two one, a two twoMeans: a rectangular table of numbers, written in brackets
- Say: a i jMeans: the entry in row i and column j: row first, then column
- Say: A transposeMeans: A with its rows and columns swapped
- Say: A inverseMeans: the matrix that undoes A; it is not one over A
- Say: the determinant of AMeans: one number telling how A scales area; if it is zero, A cannot be undone
- Say: the identity matrix IMeans: the matrix that changes nothing: 1s on the diagonal, 0s elsewhere
- Say: A x equals bMeans: a system of equations in one line: which x does A turn into b?
- Say: A v equals lambda vMeans: A only stretches v by the number lambda, an eigenvalue, without turning it
The story: A recipe for many cakes
Tap a line to see its words.
- A table for two cakes: row one is cups of flour, row two is eggs. Column one is a plain cake, column two a chocolate cake.
- The entry in row two, column two is 3: a chocolate cake needs 3 eggs.
- Today the baker makes 4 plain cakes and 2 chocolate ones.
- The table times the numbers of cakes gives the shopping list: 12 cups of flour and 10 eggs.
- The shopβs order sheet writes the same numbers with rows and columns swapped, so cakes run down the page.
- Tomorrow the store holds 16 cups of flour and 14 eggs, all to be used. How many of each cake?
- The determinant is 4, not zero, so for any stock there is exactly one answer.
- The inverse undoes the recipe; doing the recipe, then the inverse, changes nothing: the identity.
- The inverse applied to the stock gives 5 plain and 3 chocolate. Check: flour 10 + 6 = 16, eggs 5 + 9 = 14.
- One cake of each kind uses 4 cups of flour and 4 eggs: the same shape as the batch, just 4 times as big.
In words this story is 930 letters long; in symbols, 233.
Read it
What does it say?
Write it
Which notation says this?
The entry in row i and column j of A.
The unknown x is what you get by undoing A on b.
A with its rows and columns swapped.
The determinant of A is not zero.
A times its inverse is the identity.
The entry in row 3 and column 1.
A only stretches v to five times its length, without turning it.
The 2-by-2 identity matrix.
The transpose of the matrix with rows (1, 2) and (3, 4).
Two equations in one line: 2xβ + xβ = 5 and xβ β xβ = 1.
The determinant of the 2-by-2 matrix with rows (a, b) and (c, d).
Doing A twice changes nothing: A is its own inverse.
Build it
Tap the tiles in order. One tile does not belong.
A x equals b.
A v equals lambda v.
A inverse times A equals the identity.
The determinant of A is not zero.
The transpose of A transpose is A.
The entry in row two, column one is 5.
Challenges
Creative puzzles that use every sign of this chapter. Take your time; there is no clock.
1 The tick grid
Row r and column c of a 3Γ3 table of 0s and 1s are shown: a tick is a 1. Judge each sentence about this matrix A.
| β | β | ||
| β | β | ||
| β |
A tick means the entry is 1, a blank means 0. Row i, column j is a_{ij}.
Rows and columns are not interchangeable: a_{12} and a_{21} sit on opposite sides of the diagonal, so A and A^T differ whenever they disagree.
2 Matrix labels
Pair each line with what it says.
Tap a line, then its meaning.
Look-alikes carry different jobs: Aβ»ΒΉ undoes, A^T swaps, and det A = 0 warns that no undoing exists.
3 Areas multiply
This should say: scaling by AB scales area by the scaling of B and the scaling of A together. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
Doing B then A multiplies the area scalings, so the determinants multiply, they do not add.
4 The scale factor
Compute the determinant: how many times larger does this matrix make areas?
3Β·2 β 2Β·1 = 4: top-left times bottom-right, minus top-right times bottom-left.
5 Only stretched
This matrix sends the vector (0, 1) onto a longer copy of itself. By what factor?
The matrix turns (0, 1) into (0, 3): the same direction, three times as long, so Ξ» = 3.
6 Undo the recipe
Which x does this matrix turn into (5, 2)?
Start from the bottom row: xβ = 2. Then xβ + 2Β·2 = 5 gives xβ = 1.
7 When can you undo?
A matrix can be undone exactly when its determinant is not zero. Write it in symbols, then compare.
One way:
If det A = 0 the matrix flattens space and information is lost, so no inverse can restore it.
8 Mirror matrix
A matrix is symmetric when swapping rows and columns changes nothing. Write it, once with A^T and once with entries a_{ij}, then compare.
One way:
A^T = A says the whole table is unchanged; a_{ij} = a_{ji} says the same, entry by entry.
9 Almost the identity
A 3Γ3 table of 0s and 1s, one tick away from the identity. Judge each sentence about this matrix A.
| β | β | ||
| β | |||
| β |
A tick means the entry is 1, a blank means 0. Row i, column j is a_{ij}. The last statement multiplies A by the vector (0, 0, 1).
The extra 1 sits in row 1, column 3, so aββ = 1 but aββ = 0, and A is neither I nor symmetric. Yet its determinant is still 1: it slides things sideways without changing volume.
10 Only stretched, never turned
Three matrices and four vectors. A tick says the matrix only stretches that vector (perhaps by a negative factor) without turning it. Judge each sentence.
| β | β | ||
| β | β | ||
| β | β | ||
| β | β |
A tick means the rowβs vector is an eigenvector of the columnβs matrix: the matrix sends it to Ξ» times itself for some number Ξ».
A = 2I doubles everything, so every vector is an eigenvector. The swap B leaves (1, 1) alone and reverses (1, β1), but turns (1, 0) into (0, 1). C stretches the axes by different amounts, so (1, 1) is turned.
11 One table, five questions
The same table with rows (1, 2) and (3, 4) is asked five different things. Pair each line with its value.
Tap a line, then its meaning.
det: 1Β·4 β 2Β·3 = β2. Transpose: rows become columns. Times (1, 0): the first column. aββ: row two, column one. Times I: unchanged.
12 Undoing a 2Γ2 table
For A with rows (a, b) and (c, d), this should be the inverse: one over the determinant times a rearranged table. One tile is wrong. Find it, then fix it.
Tap the tile that is wrong.
The recipe swaps a and d and changes the signs of b and c. Check with the identity: multiplying A by (d, βb; βc, a) gives (ad β bc) times I.
13 The shopping list
Row one is cups of flour, row two is eggs; the columns are two kinds of bread. Today: 3 loaves of the first kind and 2 of the second. What goes on the shopping list?
Each row of the table meets the column of amounts: flour 2Β·3 + 1Β·2 = 8, eggs 1Β·3 + 3Β·2 = 9.
14 Flattened
Compute the determinant. What does the answer say about undoing this matrix?
2Β·6 β 4Β·3 = 0. The second row is one and a half times the first, so the matrix squashes the plane onto a line and has no inverse.
15 The same direction
This matrix sends (1, 1) to a longer copy of itself. By what factor?
Row one: 1Β·1 + 2Β·1 = 3; row two: 2Β·1 + 1Β·1 = 3. The result is (3, 3) = 3Β·(1, 1), so Ξ» = 3.
16 One step back
A adds the second entry onto the first. Which matrix undoes it?
To undo adding, subtract: the inverse puts β1 where A had +1. Check: A times it gives the identity. The other options are the transpose and a sign in the wrong place.
17 The matrix that does nothing
The identity changes no area and is its own inverse. Write that in symbols, then compare.
One way:
A matrix that moves nothing scales areas by 1, and undoing nothing is doing nothing again.
Chapter finished. The symbols are yours.
Tell your own
Tell a two-line story of mixing things in different amounts β paints, teas or seeds β with a table, a vector of amounts and what comes out, in signs only.
Kept on this device only.