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1. Matrices: representing data, adding and multiplying by a scalar

Warm-up

Three quick questions from earlier lessons.

Discover

A table has 2 rows and 3 columns of numbers. How many numbers does it hold?

Key idea

A matrix is a rectangular table of numbers.

m \times n

matrix has

m

rows and

n

columns; the entry in row

i

, column

j

a_{ij}

. Matrices of the same size add and subtract entry by entry, and a scalar

k

multiplies every entry:

3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & -3 \end{pmatrix}

Worked example

Add $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 5 & 0 \\ -1 & 2 \end{pmatrix}$ .

1 + 5, 2 + 0, 3 + (−1), 4 + 2

Same positions.

\begin{pmatrix} 6 & 2 \\ 2 & 6 \end{pmatrix}

Result.

Your turn, guided

Your turn: the top-left entry of 4 $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ .

8

4 · 2.

Practice ★

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

A shop records sales in a matrix with one row for each of its 4 stores and one column for each of 7 days. What are the dimensions of the matrix?
1. 4 × 7
2. 7 × 4
3. 28 × 1

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 2 & 0 \\ 6 & 2 \end{pmatrix}$ . What is the entry in row 2, column 1 of 3A?
Let A = $\begin{pmatrix} 8 & 2 \\ -1 & -3 \end{pmatrix}$ . What is the entry in row 2, column 1 of 3A?

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 4 & 6 \\ 1 & 8 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 2 \\ 5 & 9 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

Summary card

Matrices

Size is rows × columns; add entry by entry; a scalar multiplies every entry.

Watch out: rows first.

2. Multiplying matrices and its properties

Warm-up

Three quick questions from earlier lessons.

Discover

To multiply a row (2, 3) by a column (4, 5), you compute 2 · 4 + 3 · 5. What is it?

23
(8, 15)

Key idea

Each entry of AB is a row of A times a column of B.

A

m \times n

and

B

n \times p

, then

AB

m \times p

: the entry in row

i

, column

j

is row $i$ of $A$ times column $j$ of $B$ (multiply pairs, then add). The inner sizes must match. Matrix multiplication is associative and distributes over addition, but in general $AB \ne BA$ .

Worked example

Compute the top-left entry of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ .

(1, 2) and (5, 7)

Row 1 of A, column 1 of B.

1 · 5 + 2 · 7 = 19

Multiply and add.

Your turn, guided

Your turn: the top-right entry of the same product.

22

1 · 6 + 2 · 8.

Practice ★

Let A = $\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?

A is a 2 × 5 matrix and B is a 5 × 8 matrix. What are the dimensions of AB?
1. 2 × 8
2. 5 × 5
3. AB is not defined

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 8 & 8 \\ 8 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}$ . What is the entry in row 2, column 1 of AB?
Let A = $\begin{pmatrix} 1 & 8 \\ 2 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 2 & -5 \\ 5 & 6 \end{pmatrix}$ . What is the entry in row 2, column 1 of AB?

Boss problem. A stand sells pens and notebooks. The sales matrix S = $\begin{pmatrix} 9 & 5 \\ 3 & 7 \end{pmatrix}$ has rows Monday, Tuesday and columns pens, notebooks. The price vector is p = $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$ (dollars). The product Sp gives each day's income.

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} 5 & 8 \\ 0 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 1 \\ -2 & 3 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?

Summary card

Multiplying matrices

(m × n)(n × p) = m × p; entry = row · column.

Watch out: AB is usually not BA.

3. Identity, determinant and inverse of a 2 × 2 matrix

Warm-up

Three quick questions from earlier lessons.

Discover

The number 5 has the inverse 1/5 because 5 · 1/5 = 1. Which number has no inverse?

Key idea

A 2 × 2 matrix has an inverse exactly when its determinant is not 0.

The identity

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

plays the role of 1 (

AI = IA = A

) and the zero matrix the role of 0. For

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

\det A = ad - bc

. If

\det A \ne 0

A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

and

AA^{-1} = I

; if

\det A = 0

A

has no inverse.

Worked example

Invert $\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$ .

4 · 6 − 7 · 2 = 10

Determinant.

\begin{pmatrix} 6 & -7 \\ -2 & 4 \end{pmatrix}

Swap and change signs.

\begin{pmatrix} 0.6 & -0.7 \\ -0.2 & 0.4 \end{pmatrix}

Divide by 10.

Your turn, guided

Your turn: det $\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$ .

1

3 · 2 − 1 · 5.

Practice ★

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 3 \\ 5 & 4 \end{pmatrix}$ ?

For which value of k does $\begin{pmatrix} k & 3 \\ 10 & 2 \end{pmatrix}$ have no inverse?

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 5 & 3 \\ 1 & 8 \end{pmatrix}$ . What is the entry in row 1, column 1 of the inverse of A?
Let A = $\begin{pmatrix} -5 & 2 \\ 7 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of the inverse of A?

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?

Exit ticket

Three questions to check you have got it.

What is the determinant of $\begin{pmatrix} -2 & -5 \\ 8 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 3 \\ 5 & 4 \end{pmatrix}$ ?
What is the determinant of $\begin{pmatrix} 7 & 8 \\ 3 & 7 \end{pmatrix}$ ?

Summary card

Determinants and inverses

det = ad − bc; inverse = (1/det)(d, −b; −c, a) when det ≠ 0.

Watch out: the inverse is not the matrix of reciprocals.

4. Matrices as transformations: matrix times vector, determinant and area

Warm-up

Three quick questions from earlier lessons.

Discover

Doubling both coordinates of every point of a square with area 1 gives a square with area…

Key idea

A matrix moves vectors; its determinant scales areas.

Multiplying a column vector by a matrix gives a new vector:

\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}

. So a

2 \times 2

matrix is a transformation of the plane; its columns are the images of

(1, 0)

and

(0, 1)

. Areas are multiplied by

|\det A|

Worked example

Apply $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ to (3, 0) and (0, 3).

(0 · 3 − 1 · 0, 1 · 3 + 0) = (0, 3)

First point.

(−3, 0)

Second point.

a 90° counterclockwise rotation; det = 1, areas unchanged

Conclusion.

Your turn, guided

Your turn: the area scale factor of $\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$ .

6

det = 3 · 2.

Practice ★

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ?

The matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ maps the point (8, 0) to (0, 8) and (0, 8) to (−8, 0). Which transformation does it represent?
1. A rotation of 90° counterclockwise about the origin
2. A reflection in the line y = x
3. A rotation of 90° clockwise about the origin

Watch out!

Practice ★★

A shape with an area of 2 square units is transformed by the matrix $\begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$ . What is the area of the image?
A shape with an area of 8 square units is transformed by the matrix $\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}$ . What is the area of the image?

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} 7 & -1 \\ -2 & 7 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 6 \\ 1 \end{pmatrix}$ ?

Summary card

Matrices as transformations

Av moves vectors; columns are the images of (1, 0) and (0, 1); areas scale by |det|.

Watch out: the area factor is the determinant, not a sum of entries.

5. Solving systems of linear equations with matrices

Warm-up

Three quick questions from earlier lessons.

Discover

To solve 5x = 20 you multiply both sides by 1/5. What does that give?

x = 4
x = 100

Key idea

A system is one matrix equation AX = B; X = A⁻¹B.

Write the coefficients as a matrix

A

(one row per equation), the unknowns as

X = \begin{pmatrix} x \\ y \end{pmatrix}

and the constants as

B

: then

AX = B

. If

\det A \ne 0

, $X = A^{-1}B$ , the only solution.

Worked example

Solve 2x + y = 5, x + y = 3.

A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}

, det = 1

Matrix.

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

Inverse.

x = 5 − 3 = 2, y = −5 + 6 = 1

Multiply.

Your turn, guided

Your turn: in that system, what is det A?

1

2 · 1 − 1 · 1.

Practice ★

Which matrix equation represents the system 4x + 5y = 18 and 4x + 2y = 35?
1. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
2. $\begin{pmatrix} 4 & 4 \\ 5 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
3. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} 18 \\ 35 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$
Which matrix equation represents the system 9x + 9y = 13 and 2x + 4y = 10?
1. $\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$
2. $\begin{pmatrix} 9 & 2 \\ 9 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$
3. $\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} 13 \\ 10 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$

Solve the system 4x + 4y = 0 and 3x + 1y = 10 by multiplying by the inverse of the coefficient matrix. What is x?

Watch out!

Practice ★★

Solve the system 4x + 4y = 0 and 3x + 1y = 10 by multiplying by the inverse of the coefficient matrix. What is x?
Solve the system 4x + 1y = 20 and 1x + 1y = 5 by multiplying by the inverse of the coefficient matrix. What is x?

Which matrix equation represents the system 4x + 5y = 18 and 4x + 2y = 35?
1. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
2. $\begin{pmatrix} 4 & 4 \\ 5 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
3. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} 18 \\ 35 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$

Exit ticket

Three questions to check you have got it.

Which matrix equation represents the system 4x + 5y = 18 and 4x + 2y = 35?
1. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
2. $\begin{pmatrix} 4 & 4 \\ 5 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
3. $\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} 18 \\ 35 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$
Which matrix equation represents the system 9x + 9y = 13 and 2x + 4y = 10?
1. $\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$
2. $\begin{pmatrix} 9 & 2 \\ 9 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$
3. $\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} 13 \\ 10 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$
Which matrix equation represents the system 6x + 2y = 32 and 6x + 4y = 39?
1. $\begin{pmatrix} 6 & 2 \\ 6 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 32 \\ 39 \end{pmatrix}$
2. $\begin{pmatrix} 6 & 6 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 32 \\ 39 \end{pmatrix}$
3. $\begin{pmatrix} 6 & 2 \\ 6 & 4 \end{pmatrix}\begin{pmatrix} 32 \\ 39 \end{pmatrix}$ = $\begin{pmatrix} x \\ y \end{pmatrix}$

Summary card

Systems with matrices

AX = B, so X = A⁻¹B when det A ≠ 0.

Watch out: one row per equation; multiply by A⁻¹ on the left.

Answers: Matrices

1. Matrices: representing data, adding and multiplying by a scalar

$-7$
$2$

4 × 7

$18$
$-3$

$-7$

$-7$
$2$
$8$

2. Multiplying matrices and its properties

$38$
$12$

2 × 8

$28$
$-6$

$114$

$38$
$12$
$4$

3. Identity, determinant and inverse of a 2 × 2 matrix

$32$
$13$

$15$

$\frac{8}{37}$
$- \frac{2}{17}$

$32$

$32$
$13$
$25$

4. Matrices as transformations: matrix times vector, determinant and area

$-29$
$-13$

A rotation of 90° counterclockwise about the origin

$4$
$32$

$-29$

$-29$
$-13$
$41$

5. Solving systems of linear equations with matrices

$\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
$\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$

$\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$

$\begin{pmatrix} 4 & 5 \\ 4 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 18 \\ 35 \end{pmatrix}$
$\begin{pmatrix} 9 & 9 \\ 2 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 13 \\ 10 \end{pmatrix}$
$\begin{pmatrix} 6 & 2 \\ 6 & 4 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}$ = $\begin{pmatrix} 32 \\ 39 \end{pmatrix}$