M Matrix Transformations (all calculations restricted to 2x2 or 2x1 matrices)

M1

Multiplication of matrices (2x2 by 2x2 or by 2x1), and multiplication by a scalar

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

Multiplying a matrix by a scalar multiplies every entry by that number. Matrix multiplication uses row-by-column products: each entry in the result is found by multiplying corresponding entries from one row of the first matrix and one column of the second, then adding.
A product $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ .
In this specification, calculations are restricted to $2\times2$ matrices and $2\times1$ column matrices.
The order matters because matrix multiplication is generally not commutative: $AB\ne BA$ .
Examiners expect the working for individual entries to be visible, especially where negative numbers occur.

Worked example

Given $A=\begin{pmatrix}2&-1\\3&4\end{pmatrix}$ and $B=\begin{pmatrix}1&5\\-2&0\end{pmatrix}$ , work out $AB$ .

1.First row: $2(1)+(-1)(-2)=4$ and $2(5)+(-1)(0)=10$ .
2.Second row: $3(1)+4(-2)=-5$ and $3(5)+4(0)=15$ .
3.Place the four row-by-column results in their corresponding positions.

Answer: $AB=\begin{pmatrix}4&10\\-5&15\end{pmatrix}$ .

Common mistakes

Don't multiply entries in the same positions instead of forming row-by-column products.
Don't lose the negative sign in $(-1)(-2)$ or $4(-2)$ .
Don't assume $AB=BA$ and reverse the order without recalculating.

Exam tip

For a matrix product, show at least one complete row-by-column calculation before writing the resulting matrix.

Tier 1 · Easy

ORIGINAL

1

Work out

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

.

[1 mark]

Tier 2 · Standard

ORIGINAL

1

Work out

\begin{pmatrix}2&-1\\3&4\end{pmatrix}\begin{pmatrix}5\\-2\end{pmatrix}

.

[2 marks]

Tier 3 · Hard

ORIGINAL

1

Given

\begin{pmatrix}p&2\\1&q\end{pmatrix}\begin{pmatrix}3&-1\\2&4\end{pmatrix}=\begin{pmatrix}13&5\\9&11\end{pmatrix}

, work out

p

and

q

.

[4 marks]

M2

The identity matrix I (2x2 only)

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

The $2\times2$ identity matrix is $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ . It plays the same role in matrix multiplication that $1$ plays in ordinary multiplication: for every compatible matrix $A$ , $AI=IA=A$ .
Multiplying a $2\times1$ column vector by $I$ therefore leaves the represented point unchanged. The positions of the entries matter: the $1$ s lie on the main diagonal and the other entries are $0$ .
The identity matrix is not the zero matrix.
It can also appear when a transformation and its inverse are combined, or when a matrix power cycles back to the starting position.
Examiners may require the identity to be recognised, written down or used in a chain of powers.

Worked example

Let $P=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ . Show that $P^2=I$ and hence work out $P^{12}$ .

1. $P^2=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}=I$ .
2.Write $P^{12}=(P^2)^6$ .
3.Therefore $P^{12}=I^6=I$ .

Answer: $P^{12}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ .

Common mistakes

Don't write $I=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ by putting the $1$ s on the wrong diagonal.
Don't treat $I$ as the zero matrix and conclude that $AI$ is zero.
Don't expand $P^{12}$ as twelve separate multiplications instead of using $P^2=I$ .

Exam tip

When a question says ‘hence’, use the established identity relation explicitly rather than restarting the calculation.

Tier 1 · Easy

ORIGINAL

1

Write down the

2\times2

identity matrix

I

.

[1 mark]

Tier 2 · Standard

ORIGINAL

1

Let

A=\begin{pmatrix}2&-3\\5&1\end{pmatrix}

. Work out both

AI

and

IA

.

[2 marks]

Tier 3 · Hard

ORIGINAL

1

Let

P=\begin{pmatrix}0&1\\1&0\end{pmatrix}

. Show that

P^2=I

and hence work out

P^{17}

.

[3 marks]

M3

Transformations of the unit square in the x-y plane, represented by a 2x2 matrix (rotations of 90/180/270 about the origin, reflections in x=0, y=0, y=x, y=-x, enlargements centred on the origin)

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

A $2\times2$ matrix represents a transformation of the unit square in the coordinate plane.
Its first column is the image of $\begin{pmatrix}1\\0\end{pmatrix}$ and its second column is the image of $\begin{pmatrix}0\\1\end{pmatrix}$ .
Multiplying the matrix by $\begin{pmatrix}x\\y\end{pmatrix}$ gives the image of $(x,y)$ .
Required transformations are rotations of $90^\circ$ , $180^\circ$ or $270^\circ$ about the origin; reflections in $x=0$ , $y=0$ , $y=x$ or $y=-x$ ; and enlargements centred on the origin.
A full description gives the angle and centre of a rotation, adding direction for $90^\circ$ or $270^\circ$ ; the mirror line of a reflection; or the scale factor and centre of an enlargement.

The unit square and its image after a 90° anticlockwise rotation about the origin.

Worked example

The matrix $T=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ acts on the point $(3,-2)$ . Work out the image and describe geometrically the single transformation represented by $T$ .

1. $T\begin{pmatrix}3\\-2\end{pmatrix}=\begin{pmatrix}0(3)+(-1)(-2)\\1(3)+0(-2)\end{pmatrix}=\begin{pmatrix}2\\3\end{pmatrix}$ .
2.The columns show $(1,0)\mapsto(0,1)$ and $(0,1)\mapsto(-1,0)$ .
3.These images identify a rotation of $90^\circ$ anticlockwise about the origin.

Answer: The image is $(2,3)$ ; the transformation is a rotation of $90^\circ$ anticlockwise about the origin.

Common mistakes

Don't read the rows, rather than the columns, as the images of the two unit vectors.
Don't describe a $90^\circ$ rotation without stating its direction and centre.
Don't use $(x,y)T$ instead of multiplying $T\begin{pmatrix}x\\y\end{pmatrix}$ .

Exam tip

To identify an unfamiliar matrix, map the two unit vectors and then give the transformation with all defining details.

Tier 1 · Easy

ORIGINAL

1

Write down the matrix for a rotation of

90^\circ

anticlockwise about the origin.

[2 marks]

Tier 2 · Standard

ORIGINAL

1

Describe geometrically the single transformation represented by

\begin{pmatrix}-1&0\\0&1\end{pmatrix}

and work out the image of

(4,-3)

.

[3 marks]

Tier 3 · Hard

ORIGINAL

1

A transformation maps

\begin{pmatrix}1\\0\end{pmatrix}

to

\begin{pmatrix}0\\-1\end{pmatrix}

and

\begin{pmatrix}0\\1\end{pmatrix}

to

\begin{pmatrix}-1\\0\end{pmatrix}

. Write its matrix, describe geometrically the single transformation represented by this matrix, and work out the image of

(3,-2)

.

[4 marks]

M4

Combination of transformations using matrix multiplication

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

Successive matrix transformations combine through matrix multiplication. If transformation $A$ acts first and transformation $B$ acts second, a vector $\mathbf{x}$ becomes $B(A\mathbf{x})=(BA)\mathbf{x}$ , so the combined matrix is $BA$ .
The right-hand matrix therefore acts first. Order is essential because $AB$ and $BA$ are generally different transformations.
After multiplying, the combined matrix may match a familiar rotation, reflection or enlargement and should be described fully when requested.
Calculations remain within $2\times2$ and $2\times1$ matrices; $\mathbf{i}$ and $\mathbf{j}$ notation is not required.
Examiners expect the action order to be translated into the correct matrix order before row-by-column multiplication begins.

Transformation A acts first and transformation B acts second, so the combined matrix is BA.

Worked example

A reflection in the line $y=x$ is followed by a rotation of $90^\circ$ clockwise about the origin. Work out the combined matrix and describe geometrically the single transformation it represents.

1.Use $S=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ for the reflection and $R=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ for the rotation.
2.Since $S$ acts first, calculate $RS=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ .
3.The matrix leaves $x$ unchanged and changes $y$ to $-y$ .

Answer: The combined transformation is reflection in the line $y=0$ .

Common mistakes

Don't write $SR$ because the transformations are copied in the order stated, even though $S$ acts first.
Don't multiply the matrices entry by entry rather than row by column.
Don't call the combined matrix a reflection without identifying the mirror line $y=0$ .

Exam tip

Write a short action chain such as $\mathbf{x}\to A\mathbf{x}\to BA\mathbf{x}$ before forming the combined matrix.

Tier 1 · Easy

ORIGINAL

1

A reflection in the

x

-axis is followed by a reflection in the

y

-axis. Work out the combined matrix.

[2 marks]

Tier 2 · Standard

ORIGINAL

1

Matrix

A

rotates points

90^\circ

anticlockwise about the origin. Matrix

B

reflects points in the

x

-axis. Transformation

A

acts before transformation

B

. Work out the combined matrix and describe its effect.

[3 marks]

Tier 3 · Hard

ORIGINAL

1

A point is reflected in

y=x

, enlarged by scale factor

3

about the origin, then rotated

90^\circ

clockwise about the origin. Work out the combined matrix, describe its geometric effect, and work out the image of

(-2,1)

.

[4 marks]

Multiplication of matrices (2x2 by 2x2 or by 2x1), and multiplication by a scalar

Tier 1 · Easy

Tier 2 · Standard

Tier 3 · Hard

The identity matrix I (2x2 only)

Tier 1 · Easy

Tier 2 · Standard

Tier 3 · Hard

Transformations of the unit square in the x-y plane, represented by a 2x2 matrix (rotations of 90/180/270 about the origin, reflections in x=0, y=0, y=x, y=-x, enlargements centred on the origin)

Tier 1 · Easy

Tier 2 · Standard

Tier 3 · Hard

Combination of transformations using matrix multiplication

Tier 1 · Easy

Tier 2 · Standard

Tier 3 · Hard

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