Matrices
Introduced| English | Meaning |
|---|---|
| matrix/ˈmeɪtrɪks/ | A rectangular block of numbers. |
| transformation/trænsfɔːˈmeɪʃn/ | A rule or operation that maps an object, set of values or distribution to another form. |
| identity matrix/aɪˈdentɪti ˈmeɪtrɪks/ | A square matrix with ones on its main diagonal and zeros elsewhere, which leaves compatible matrices unchanged under multiplication. |
| determinant/dɪˈtɜːmɪnənt/ | A scalar calculated from a square matrix that indicates properties such as invertibility and area or volume scaling. |
| inverse/ɪnˈvɜːs/ | An operation or mathematical object that reverses the effect of another under the specified combination rule. |
| singular/ˈsɪŋɡjʊlə/ | singular |
| invariant/ɪnˈveərɪənt/ | invariant |
| enlargement/enˈlɑːdʒmənt/ | A transformation that scales distances from a fixed centre by a specified scale factor. |
The mathematics that powers computer graphics
- Every time you rotate an image on your phone, a matrix 矩阵 multiplication is doing the work behind the scenes.
- Matrices are the language of transformations 变换 — they describe rotations, reflections, stretches, and projections in a single compact object.
Matrix basics
- A matrix is a rectangular block of numbers. Multiply row × column.
- The identity matrix 单位矩阵 $I$ leaves a matrix unchanged: $AI = IA = A$.
Worked example. $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 1(5) + 2(6) \\ 3(5) + 4(6) \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$.
Matrices
(x', y') = M(x, y)
A matrix transforms the plane; its determinant is the area scale factor (negative = a flip).
Multiply [[1, 2], [3, 4]] by [[5], [6]]. The top entry of the result is?
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1(5) + 2(6) = 5 + 12 = 17.
Put the steps for finding the inverse of a 2x2 matrix [[a,b],[c,d]] in order.
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The inverse is (1/det)[[d,−b],[−c,a]] — swap the diagonal, negate the off-diagonal, divide by det.
Determinants 行列式 and inverses 逆
- For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$: the determinant is $\det A = ad - bc$.
- If $\det A \neq 0$ (non-singular 奇异), the inverse is $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
Singular matrices have no inverse. If $\det A = 0$, the matrix is singular — it squashes the plane into a line, losing information that can't be recovered.
What is the determinant of the matrix [[3, 1], [2, 4]] (ad − bc)?
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det = 3×4 − 1×2 = 12 − 2 = 10.
A 2×2 matrix has an inverse exactly when its determinant is:
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A non-singular matrix (det ≠ 0) is invertible; det = 0 means singular (no inverse).
For A = [[3, 1], [2, 4]] with det = 10, the top-left entry of A⁻¹ is d/det. Find it.
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A⁻¹ = (1/10)[[4, -1], [-2, 3]]. Top-left = 4/10 = 0.4.
Matrices as transformations
- A $2\times2$ matrix represents a transformation of the plane; $\det A$ = the area scale factor.
A 2x2 matrix maps the unit square to a parallelogram; its determinant is the area scale factor.
- A product $AB$ means "do $B$, then $A$". Points/lines that don't move are invariant 不变.
For a transformation matrix, the determinant gives the:
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The determinant is the factor by which areas are scaled by the transformation.
For matrices A and B, the product AB means “do A first, then B”.
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False. AB acts right to left: AB(x) = A(B(x)), so B is applied first and then A.
Common transformations
| Matrix | Transformation |
|---|---|
| $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ | Reflection in $y$-axis |
| $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | Rotation $90°$ anticlockwise |
| $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ | Enlargement 放大 scale factor $k$ |
- Know matrix addition, the zero matrix, the identity (or unit) matrix, and invariant points and invariant lines of a transformation.
You've got it
- $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$; invertible when $\det \neq 0$
- $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
- a matrix is a transformation; $\det$ = area scale factor; $AB$ = "do $B$, then $A$"
Write from a determinant of 13
- The matrix has rows (5, 2) and (1, 3).
- Find its determinant. A determinant of 0 means there is no inverse.
A matrix has rows (5, 2) and (1, 3). What is its determinant, and when does a square matrix have no inverse?
For a 2 by 2 matrix the determinant is ad minus bc. A non-zero determinant is what makes the inverse exist.
What is the determinant?
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5 times 3 minus 2 is 13.