International A LevelFurther MathsEdexcelFurther Pure 1Revision NotesMatricesMatrix AlgebraMultiplying Matrices

Multiplying Matrices (Edexcel International A Level Further Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Multiplying Matrices

How do I multiply a 2x2 matrix by a 2x1 matrix?

  • Multiply the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix, writing their sum in the answer matrix

  • The answer will be a 2 × 1 matrix

    • open parentheses table row a b row c d end table close parentheses open parentheses table row x row y end table close parentheses equals open parentheses table row cell a x plus b y end cell row cell c x plus d y end cell end table close parentheses

  • open parentheses table row 1 2 row 3 4 end table close parentheses open parentheses table row 10 row 20 end table close parentheses equals open parentheses table row cell 1 cross times 10 plus 2 cross times 20 end cell row cell 3 cross times 10 plus 4 cross times 20 end cell end table close parentheses equals open parentheses table row cell 10 plus 40 end cell row cell 30 plus 80 end cell end table close parentheses equals open parentheses table row 50 row 110 end table close parentheses

How do I multiply a 2x2 matrix by another 2x2 matrix?

  • Multiply the corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix, writing their sum in the answer matrix 

  • The answer will be a 2 × 2 matrix

    • open parentheses table row a b row c d end table close parentheses open parentheses table row A B row C D end table close parentheses equals open parentheses table row cell a A plus b C end cell cell a B plus b D end cell row cell c A plus d C end cell cell c B plus d D end cell end table close parentheses

  • open parentheses table row 1 2 row 3 4 end table close parentheses open parentheses table row 5 10 row 20 25 end table close parentheses equals open parentheses table row cell 1 cross times 5 plus 2 cross times 20 end cell cell 1 cross times 10 plus 2 cross times 25 end cell row cell 3 cross times 5 plus 4 cross times 20 end cell cell 3 cross times 10 plus 4 cross times 25 end cell end table close parentheses equals open parentheses table row cell 5 plus 40 end cell cell 10 plus 50 end cell row cell 15 plus 80 end cell cell 30 plus 100 end cell end table close parentheses equals open parentheses table row 45 60 row 95 130 end table close parentheses

  • This process becomes more natural the more times you do it!

How do I square a matrix?

  • Do not square each individual element inside the matrix

  • Write out a matrix multiplication

    • If bold P equals open parentheses table row 2 4 row 1 cell negative 3 end cell end table close parentheses then bold P squared equals bold P cross times bold P equals open parentheses table row 2 4 row 1 cell negative 3 end cell end table close parentheses open parentheses table row 2 4 row 1 cell negative 3 end cell end table close parentheses equals open parentheses table row cell 2 cross times 2 plus 4 cross times 1 end cell cell 2 cross times 4 plus 4 cross times open parentheses negative 3 close parentheses end cell row cell 1 cross times 2 plus open parentheses negative 3 close parentheses cross times 1 end cell cell 1 cross times 4 plus open parentheses negative 3 close parentheses cross times open parentheses negative 3 close parentheses end cell end table close parentheses equals open parentheses table row 8 cell negative 4 end cell row cell negative 1 end cell 13 end table close parentheses

  • It is possible to have negative elements in a squared matrix

How do I multiply matrices of any dimensions?

  • To multiply a matrix by another matrix:

    • The number of columns in the first matrix must be equal to the number of rows in the second matrix

    • For example, the first matrix is m cross times n and the second matrix is n cross times p

    • The order of the resultant matrix will be m cross times p

  • Multiply corresponding elements in the row of the first matrix with the corresponding elements in the column of the second matrix

    • Place their sums in the resultant matrix

      • For example, if bold A equals open parentheses table row a b c row d e f end table close parentheses, bold B equals open parentheses table row g h row i j row k l end table close parentheses

      • then Error converting from MathML to accessible text. 

      • whereas bold BA equals open parentheses table row cell left parenthesis g a plus h d right parenthesis end cell cell left parenthesis g b plus h e right parenthesis end cell cell left parenthesis g c plus h f right parenthesis end cell row cell left parenthesis i a plus j d right parenthesis end cell cell left parenthesis i b plus j e right parenthesis end cell cell left parenthesis i c plus j f right parenthesis end cell row cell left parenthesis k a plus l d right parenthesis end cell cell left parenthesis k b plus l e right parenthesis end cell cell left parenthesis k c plus l f right parenthesis end cell end table close parentheses

  • It is possible for AB to exist but BA not to exist

    • For square matrices of the same order, AB and BA will both exist

What does commutative mean?

  • Commutative means "swapping the order doesn't change the result"

    • 5 × 4 = 4 × 5 and 3 + 2 = 2 + 3

      • Multiplication and addition of numbers are commutative

    • 4 ÷ 2 ≠ 2 ÷ 4 and 5 - 3 ≠ 3 - 5

      • Division and subtraction of numbers are not commutative

  • However, matrix multiplication is not commutative

    • In general AB ≠ BA

  • For example, open parentheses table row 1 2 row 3 4 end table close parentheses open parentheses table row 0 1 row 5 1 end table close parentheses equals open parentheses table row 10 3 row 20 7 end table close parentheses but open parentheses table row 0 1 row 5 1 end table close parentheses open parentheses table row 1 2 row 3 4 end table close parentheses equals open parentheses table row 3 4 row 8 14 end table close parentheses

What does associative mean?

  • Associative means "it doesn't matter which order you group operations into"

    • To do 5 + 4 + 3, either (5 + 4) + 3 or 5 + (4 + 3) works

    • To do 8 × 9 × 10, either (8 × 9) × 10 or 8 × (9 × 10) works 

      • Multiplication and addition of numbers are associative

    • (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2) and (5 - 4) - 3 ≠ 5 - (4 - 3)

      • Division and subtraction of numbers are not associative

  • Matrix multiplication is associative

    • (AB)C ≡ A(BC)

  • To multiply three matrices together, it is fine to either:

    • start by multiplying the first two together,

    • or start by multiplying the second two together

      • Just don't switch the order

      • A(BC) is not the same as (BC)A

Worked Example

If bold P equals open parentheses table row 3 1 row cell negative 2 end cell 0 end table close parentheses, bold Q equals open parentheses table row 5 cell negative 5 end cell row 4 2 end table close parentheses and bold R equals open parentheses table row 10 cell negative 1 end cell 4 row 8 0 5 end table close parentheses, find the following:

(a) bold PQ 

Write out bold PQ in full

open parentheses table row 3 1 row cell negative 2 end cell 0 end table close parentheses cross times open parentheses table row 5 cell negative 5 end cell row 4 2 end table close parentheses

Multiply the matrices

open parentheses table row cell open parentheses 3 cross times 5 space plus space 1 cross times 4 close parentheses end cell cell open parentheses 3 cross times negative 5 space plus space 1 cross times 2 close parentheses end cell row cell open parentheses negative 2 cross times 5 space plus space 0 cross times 4 close parentheses end cell cell open parentheses negative 2 cross times negative 5 space plus space 0 cross times 2 close parentheses end cell end table close parentheses equals open parentheses table row cell open parentheses 15 space plus space 4 close parentheses end cell cell open parentheses negative 15 space plus space 2 close parentheses end cell row cell open parentheses negative 10 space plus space 0 close parentheses end cell cell open parentheses 10 space plus space 0 close parentheses end cell end table close parentheses

Simplify

bold PQ equals open parentheses table row 19 cell negative 13 end cell row cell negative 10 end cell 10 end table close parentheses

(b) bold PR

Write out bold PR in full

open parentheses table row 3 1 row cell negative 2 end cell 0 end table close parentheses cross times open parentheses table row 10 cell negative 1 end cell 4 row 8 0 5 end table close parentheses

The order (dimensions) agree, as bold P has 2 columns and bold R has 2 rows
Multiply the matrices

open parentheses table row cell open parentheses 3 cross times 10 space plus space 1 cross times 8 close parentheses end cell cell open parentheses 3 cross times negative 1 space plus space 1 cross times 0 close parentheses end cell cell open parentheses 3 cross times 4 space plus space 1 cross times 5 close parentheses end cell row cell open parentheses negative 2 cross times 10 space plus space 0 cross times 8 close parentheses end cell cell open parentheses negative 2 cross times negative 1 space plus space 0 cross times 0 close parentheses end cell cell open parentheses negative 2 cross times 4 space plus space 0 cross times 5 close parentheses end cell end table close parentheses

Simplify

bold PR equals open parentheses table row 38 cell negative 3 end cell 17 row cell negative 20 end cell 2 cell negative 8 end cell end table close parentheses

(c) bold Q squared

Write out bold Q squared as bold Q cross times bold Q

open parentheses table row 5 cell negative 5 end cell row 4 2 end table close parentheses cross times open parentheses table row 5 cell negative 5 end cell row 4 2 end table close parentheses

Multiply the matrices

open parentheses table row cell open parentheses 5 cross times 5 space plus space minus 5 cross times 4 close parentheses end cell cell open parentheses 5 cross times negative 5 space plus space minus 5 cross times 2 close parentheses end cell row cell open parentheses 4 cross times 5 space plus space 2 cross times 4 close parentheses end cell cell open parentheses 4 cross times negative 5 space plus space 2 cross times 2 close parentheses end cell end table close parentheses equals open parentheses table row cell open parentheses 25 space plus space minus 20 close parentheses end cell cell open parentheses negative 25 space plus space minus 10 close parentheses end cell row cell open parentheses 20 space plus space 8 close parentheses end cell cell open parentheses negative 20 space plus space 4 close parentheses end cell end table close parentheses

Simplify

bold Q squared equals open parentheses table row 5 cell negative 35 end cell row 28 cell negative 16 end cell end table close parentheses

(d) Explain why the matrix bold RP does not exist.

An m cross times n matrix can only be multiplied by an n cross times p matrix

The order (dimensions) do not agree, as bold R has 3 columns but bold P has 2 rows

The Identity Matrix

What is the Identity Matrix?

  • The identity matrix, bold I, is a square matrix with:

    • Ones along the leading diagonal (from top-left to bottom-right)

    • and zeros everywhere else

      • The 2 × 2 identity matrix is bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses

      • The 3 × 3 identity matrix is bold I equals open parentheses table row 1 0 0 row 0 1 0 row 0 0 1 end table close parentheses

    • The notation bold I subscript n can be used to specify the n cross times n identity matrix

  • Multiplying any square matrix by the same-sized identity matrix leaves it unchanged

    • Both bold AI equals bold A and bold IA equals bold A

    • open parentheses table row a b row c d end table close parentheses open parentheses table row 1 0 row 0 1 end table close parentheses equals open parentheses table row a b row c d end table close parentheses and open parentheses table row 1 0 row 0 1 end table close parentheses open parentheses table row a b row c d end table close parentheses equals open parentheses table row a b row c d end table close parentheses

      • This result can be proved by multiplying together the matrices

Examiner Tips and Tricks

The identity matrix is an important matrix which you should know or recognise as bold I in a question.

Worked Example

If bold A equals open parentheses table row 0 2 row 2 0 end table close parentheses show that bold A squared equals 4 bold I.

Write out bold A squared as bold A cross times bold A

open parentheses table row 0 2 row 2 0 end table close parentheses cross times open parentheses table row 0 2 row 2 0 end table close parentheses

Multiply the matrices

table row cell open parentheses table row 0 2 row 2 0 end table close parentheses cross times open parentheses table row 0 2 row 2 0 end table close parentheses end cell equals cell open parentheses table row cell open parentheses 0 cross times 0 space plus space 2 cross times 2 close parentheses end cell cell open parentheses 0 cross times 2 space plus space 2 cross times 0 close parentheses end cell row cell open parentheses 2 cross times 0 space plus space 0 cross times 2 close parentheses end cell cell open parentheses 2 cross times 2 space plus space 0 cross times 0 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 0 row 0 4 end table close parentheses end cell end table

Write in terms of the identity matrix, bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses by factorising out 4

stretchy left parenthesis table row 4 0 row 0 4 end table stretchy right parenthesis equals 4 stretchy left parenthesis table row 1 0 row 0 1 end table stretchy right parenthesis equals 4 bold I

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    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.