Matrices
Introduced| English | Meaning |
|---|---|
| matrix/ˈmeɪtrɪks/ | A rectangular array of numbers arranged in rows and columns, with multiplication defined by row–column products. |
| inverse/ɪnˈvɜːs/ | An operation or mathematical object that reverses the effect of another under the specified combination rule. |
| zero matrix/ˈzɪərəʊ ˈmeɪtrɪks/ | The matrix of all zeros, the additive identity of matrices. |
| singular/ˈsɪŋɡjʊlə/ | For a square matrix, having zero determinant and therefore no inverse. |
| 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. |
| minor/ˈmaɪnə/ | The determinant left after deleting one specified row and column of a square matrix. |
| cofactor/ˈkəʊˌfæktə/ | A minor multiplied by the sign of its position, (−1) raised to the row number plus the column number. |
| transformation/trænsfɔːˈmeɪʃn/ | A rule or operation that maps an object, set of values or distribution to another form. |
| enlargement/enˈlɑːdʒmənt/ | A transformation that scales distances from a fixed centre by a specified scale factor. |
| invariant/ɪnˈveərɪənt/ | Unchanged under a transformation: an invariant point keeps its coordinates; an invariant line maps onto itself, although points may move along it. |
Undo an image edit
- A graphics program can store a rotation or stretch as a matrix 矩阵. Its columns show where the two unit directions move; multiplying maps every point in the image.
- To undo the edit, an inverse 逆 must exist. To combine edits, their order matters. A correct answer should explain both the calculation and what happens to the shape.
Check the sizes before calculating
- A matrix with $m$ rows and $n$ columns has size $m\times n$. Add or subtract only matrices of the same size, entry by entry; the zero matrix 零矩阵 has every entry zero.
- If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$. Match the inner sizes, then multiply each row by each column. A product in the reverse order may have a different size or may not exist.
For the non-square matrices
For example, entry $(1,2)$ is $1(0)+2(-1)+(-1)(2)=-4$. The product $BA$ exists but is $3\times3$, so it cannot equal $AB$.
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.
Let A=[[2,−1,3],[1,0,2]], C=[[0,1,−2],[3,−1,1]] and B=[[1,2],[0,−1],[2,1]]. Find A+C, A−C, AB and BA. State their sizes and explain why A+B is undefined.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Identity, order and singular 奇异 matrices
- The identity matrix 单位矩阵 $I_n$ is $n\times n$, with ones on its main diagonal and zeros elsewhere. For $A$ of size $m\times n$, $I_mA=A=AI_n$. Usually $AB\ne BA$; matrix multiplication is not commutative.
- A square matrix is singular when its determinant 行列式 is zero; it has no inverse. A non-singular square matrix has a unique inverse with $AA^{-1}=A^{-1}A=I$. A singular plane map may collapse the plane to a line or to a single point.
For $A=\begin{pmatrix}3&1\\2&4\end{pmatrix}$,
For any $2\times2$ matrix, swap the two diagonal entries, negate the other two entries, then divide by $ad-bc$, provided this determinant is non-zero.
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.
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.
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.
Expand a three by three determinant
- To find a $3\times3$ determinant, choose a row and delete the row and column of each chosen entry. Each remaining $2\times2$ determinant is a minor 余子式. Across the first row the signs are $+,-,+$.
- For a parameter, factor the determinant and state every value making it zero. Do not divide by a parameter expression before checking whether it can vanish.
Expanding across the first row,
The middle minus sign belongs to the position, even if the minor itself is negative.
For K=[[t,1,0],[1,t,0],[0,0,3]], find every real t for which K is singular. For other t, find K⁻¹. What do you check before dividing?
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Find and check a three by three inverse
- A cofactor 代数余子式 is a minor multiplied by its position sign. Use the sign pattern $\begin{pmatrix}+&-&+\\-&+&-\\+&-&+\end{pmatrix}$, then transpose the cofactor matrix: rows become columns. Divide that transpose by the non-zero determinant.
- Check the result by multiplication. Computing one cofactor incorrectly or forgetting the transpose can give a plausible-looking matrix that is not an inverse.
For the matrix $M$ above, the cofactor matrix and inverse are
For example, the second row of $M$ times the first column of $M^{-1}$ gives $(0+2-2)/5=0$, as required for entry $(2,1)$ of $I_3$. Check all nine entries.
Find the determinant and inverse of N=[[1,0,2],[1,1,0],[0,2,1]]. Show a cofactor matrix and verify your inverse by multiplication.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Undo several operations in reverse order
- For non-singular square matrices of the same size, $(AB)^{-1}=B^{-1}A^{-1}$. Multiplication is associative, so $ABB^{-1}A^{-1}=I$. The reverse order removes the last operation first.
- Extend this to three factors: $(ABC)^{-1}=C^{-1}B^{-1}A^{-1}$. In a matrix equation, multiply on the correct side: from $AXB=C$, obtain $X=A^{-1}CB^{-1}$ when both inverses exist.
There is no general cancellation rule that moves a matrix through another matrix. Also, $AB$ can be square and singular even when $A$ and $B$ are non-square: calculate the product before trying to take its determinant.
Let U=[[1,0],[0,1],[1,1]] and V=[[2,1,0],[0,1,3]]. Calculate UV and VU. Prove UV is singular and decide whether VU is singular. Explain why det(U)det(V) is not a valid calculation.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Let A=[[1,1],[0,1]], B=[[2,0],[0,1]] and C=[[0,−1],[1,0]]. Find ABC and its inverse. Also prove the inverse-product rule for three non-singular matrices without assuming they commute.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Solve AXB=C where A=[[1,1],[0,1]], B=[[2,0],[0,1]] and C=[[4,3],[2,1]]. State which side each inverse multiplies on and verify the result.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Build a plane transformation 变换 from columns
- For column coordinates, $\binom XY=T\binom xy$. The first column is the image of $(1,0)$ and the second is the image of $(0,1)$. These determine a linear transformation; the origin stays fixed.
- Recognise rotation, reflection, enlargement 放大, stretch and shear. State the centre or axis, direction and factor as appropriate. A translation with a non-zero displacement cannot be represented by a $2\times2$ matrix acting on these column coordinates.
| Transformation | Matrix |
|---|---|
| Anticlockwise rotation through $\theta$ about the origin | $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ |
| Reflection in the $x$-axis | $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ |
| Reflection in $y=x$ | $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ |
| Enlargement, centre origin, scale factor $k$ | $\begin{pmatrix}k&0\\0&k\end{pmatrix}$ |
| Stretch parallel to the $x$-axis, factor $k$ | $\begin{pmatrix}k&0\\0&1\end{pmatrix}$ |
| Shear fixing the $x$-axis, $(0,1)\mapsto(h,1)$ | $\begin{pmatrix}1&h\\0&1\end{pmatrix}$ |
A linear plane transformation sends (1,0) to (2,−1) and (0,1) to (−1,3). Find its matrix and the image of (3,2). Could this matrix represent a non-zero translation? Explain.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Describe H=[[1,−3],[0,1]] fully, find H⁻¹, and find all of its invariant points. Does determinant 1 mean H is the identity?
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Compose edits and interpret area
- The product $AB$ applies $B$ first, then $A$. A stretch $S=\begin{pmatrix}2&0\\0&1\end{pmatrix}$ followed by a $90^\circ$ anticlockwise rotation $R$ gives $RS=\begin{pmatrix}0&-1\\2&0\end{pmatrix}$. Reversing them gives $SR=\begin{pmatrix}0&-2\\1&0\end{pmatrix}$.
- The area scale factor is $|\det T|$. The sign of $\det T$ tells whether orientation is reversed, not whether an area is negative. A negative enlargement factor reverses both coordinate directions, so its determinant $k^2$ is positive.
Both images have area 2. Their determinants have opposite signs: area magnitude and orientation are different properties.
The two basis vectors land on the columns of the matrix. The magnitude of the determinant measures area change.
Matrices
(x', y') = M(x, y)
A matrix transforms column coordinates. The area scale factor is the absolute value of its determinant; a negative determinant reverses orientation.
The area scale factor of a plane transformation is:
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Area is multiplied by |det T|. The sign of det T records orientation.
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.
Stretch parallel to the x-axis by factor 3, then reflect in y=x. Find the composite matrix. Compare the result with the reversed order by transforming (1,2). Give full descriptions of both individual transformations.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
T=[[-2,0],[0,3]] maps a triangle of area 5 onto another triangle. Find its image area, state whether orientation changes, and describe a sequence of simple transformations giving T. Compare this with enlargement scale factor −2.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Find points that stay fixed
- An invariant 不变 point keeps its own coordinates: $T\binom xy=\binom xy$. Solve $(T-I)\binom xy=\binom00$. The origin is always a solution; a line of fixed points requires more solutions than the origin alone.
- For $T=\begin{pmatrix}2&1\\0&1\end{pmatrix}$, the equations reduce to $x+y=0$. Every point on $y=-x$ stays fixed. If $\det(T-I)\ne0$, the origin is the only invariant point; if it is zero, solve the equations to find the actual set.
For $T_\theta=\begin{pmatrix}1&1\\0&1\end{pmatrix}R_\theta$, a non-zero fixed point requires
On $0<\theta<2\pi$, put $t=\tan(\theta/2)$, checking $\theta=\pi$ separately. Then $4t^2-2t=0$, giving $\theta=2\arctan(1/2)$. Substitution gives $T_\theta-I=\frac15\begin{pmatrix}2&-1\\4&-2\end{pmatrix}$, so the fixed line is $y=2x$. The determinant condition alone does not give its equation.
For T=[[k,2],[1,2]], find the value of k giving a line of invariant points, and find that line. Check that T is non-singular at your value. Why is a determinant condition alone not the complete answer?
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
A line can stay invariant while its points move
- An invariant line maps onto itself as a set. Its individual points may move along it. For the non-singular map $\binom XY=\begin{pmatrix}a&b\\c&d\end{pmatrix}\binom xy$, put $y=mx$ and require $Y=mX$ for every point on that line.
- This gives $bm^2+(a-d)m-c=0$. State equations of lines, not just slopes, and test the vertical line $x=0$ separately: assuming a finite gradient can miss it.
For $T=\begin{pmatrix}2&0\\1&3\end{pmatrix}$ the finite-gradient equation is $-m-1=0$, so $y=-x$ is invariant. The vertical line $x=0$ is also invariant, because $(0,y)\mapsto(0,3y)$.
No non-zero point on $x=0$ stays fixed. For this matrix, solving $(T-I)(x,y)^{\mathsf T}=0$ gives only the origin: invariant lines and lines of invariant points answer different questions.
For T=[[3,2],[1,2]], find every invariant line through the origin. Which of these is a line of invariant points? Test the vertical line separately.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Find all invariant lines through the origin for T=[[3,0],[2,1]], and all invariant points. Explain what a method using only finite gradients would miss.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
A90° anticlockwise rotation about the origin has matrix R=[[0,−1],[1,0]]. Find its invariant points and decide whether any real straight line through the origin is invariant. Show the algebra and check the vertical case.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Check lines away from the origin
- For a line $y=mx+c$, substitute its general point into $(X,Y)$ and require $Y=mX+c$ for all $x$. Compare the coefficient of $x$ and the constant term. Test vertical lines $x=c$ separately.
- For the shear $H=\begin{pmatrix}1&2\\0&1\end{pmatrix}$, every horizontal line $y=c$ is invariant: $(x,c)\mapsto(x+2c,c)$. Only $y=0$ is a line of invariant points. On $y=1$, every point moves two units right.
The two marked points move along y=1, so the line stays invariant. Every point on y=0 stays in its own position.
For H=[[1,−1],[0,1]], find every invariant straight line, including lines not through the origin. Identify its lines of invariant points and describe the motion on y=2.
Write the defining equations and restrictions before calculating, then verify the resulting matrix or geometric set.
Choose the condition before the algebra
- First identify whether the task asks for a product, an inverse, a transformation, an invariant point or an invariant line. State sizes and restrictions before calculating, and describe geometric transformations fully.
- Verify an inverse by a product, a fixed point by $T\mathbf v=\mathbf v$, and an invariant line by its equation after transformation. A numerical check of one point cannot prove that an entire line is invariant.
Check sizes and multiplication order first. Use a non-zero determinant for an inverse, and distinguish fixed points from a line that maps onto itself.
A matrix has rows (5,2) and (1,3). What is its determinant?
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5 times 3 minus 2 is 13.