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Multiplying Matrices (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Multiplying matrices

How do I multiply two matrices?

  • To multiply two matrices

    • the number of columns in the first matrix

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

  • For example, consider bold A equals open parentheses table row 1 3 row 7 4 end table close parentheses and bold B equals open parentheses table row 1 0 5 row 8 2 9 end table close parentheses

    • You can multiply bold A cross times bold B

      • because the number of columns in bold A (2) matches the number of rows in bold B (2)

    • But you cannot multiply bold B cross times bold A

      • because the number of columns in bold B (3) does not match the number of rows in bold A (2)

  • Multiplying matrices involves

    • multiplying the corresponding elements in a row of the first matrix

      • with the corresponding elements in a column of the second matrix

      • and writing the sum of the products in the answer matrix

    • It is easiest to see this through some examples

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

How do I multiply a 2×2 matrix by a 2×1 matrix?

  • The answer will be a 2 x 1 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

    • 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 2×2 matrix by another 2×2 matrix?

  • The answer will be a 2 x 2 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 

    • 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

How do I multiply a 2×2 matrix by a 2×3 matrix?

  • The answer will be a 2 x 3 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 

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

  • table row cell open parentheses table row 1 3 row 7 4 end table close parentheses open parentheses table row 1 0 5 row 8 2 9 end table close parentheses end cell equals cell open parentheses table row cell 1 cross times 1 plus 3 cross times 8 end cell cell 1 cross times 0 plus 3 cross times 2 end cell row cell 7 cross times 1 plus 4 cross times 8 end cell cell 7 cross times 0 plus 4 cross times 2 end cell end table table row cell 1 cross times 5 plus 3 cross times 9 end cell row cell 7 cross times 5 plus 4 cross times 9 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 plus 24 end cell cell 0 plus 6 end cell row cell 7 plus 32 end cell cell 0 plus 8 end cell end table table row cell 5 plus 27 end cell row cell 35 plus 36 end cell end table close parentheses end cell row blank equals cell open parentheses table row 25 6 row 39 8 end table table row 32 row 71 end table close parentheses end cell end table

How do I multiply a 3×3 matrix by another 3×3 matrix?

  • The answer will be a 3 x 3 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 

    • open parentheses table row a b c row d e f row g h i end table close parentheses open parentheses table row A B C row D E F row G H I end table close parentheses equals open parentheses table row cell a A plus b D plus c G end cell cell a B plus b E plus c H end cell cell a C plus b F plus c I end cell row cell d A plus e D plus f G end cell cell d B plus e E plus f H end cell cell d C plus e F plus f I end cell row cell g A plus h D plus i G end cell cell g B plus h E plus i H end cell cell g C plus h F plus i I end cell end table close parentheses

  • table row cell open parentheses table row 1 0 3 row 0 cell negative 5 end cell 0 row 2 7 1 end table close parentheses open parentheses table row 0 2 3 row cell negative 1 end cell 0 cell negative 2 end cell row 4 0 6 end table close parentheses end cell equals cell open parentheses table row cell 1 cross times 0 plus 0 cross times open parentheses negative 1 close parentheses plus 3 cross times 4 end cell cell 1 cross times 2 plus 0 cross times 0 plus 3 cross times 0 end cell cell 1 cross times 3 plus 0 cross times open parentheses negative 2 close parentheses plus 3 cross times 6 end cell row cell 0 cross times 0 plus open parentheses negative 5 close parentheses cross times open parentheses negative 1 close parentheses plus 0 cross times 4 end cell cell 0 cross times 2 plus open parentheses negative 5 close parentheses cross times 0 plus 0 cross times 0 end cell cell 0 cross times 3 plus open parentheses negative 5 close parentheses cross times open parentheses negative 2 close parentheses plus 0 cross times 6 end cell row cell 2 cross times 0 plus 7 cross times open parentheses negative 1 close parentheses plus 1 cross times 4 end cell cell 2 cross times 2 plus 7 cross times 0 plus 1 cross times 0 end cell cell 2 cross times 3 plus 7 cross times open parentheses negative 2 close parentheses plus 1 cross times 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 0 plus 0 plus 12 end cell cell 2 plus 0 plus 0 end cell cell 3 plus 0 plus 18 end cell row cell 0 plus 5 plus 0 end cell cell 0 plus 0 plus 0 end cell cell 0 plus 10 plus 0 end cell row cell 0 minus 7 plus 4 end cell cell 4 plus 0 plus 0 end cell cell 6 minus 14 plus 6 end cell end table close parentheses end cell row blank equals cell open parentheses table row 12 2 21 row 5 0 10 row cell negative 3 end cell 4 cell negative 2 end cell end table close parentheses end cell end table

How do I square a matrix?

  • Only square matrices (2 x 2 or 3 x 3) can be squared

  • Do not square each individual element

  • Write out a matrix multiplication

    • E.g. 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 after squaring a matrix

When multiplying, does it matter which matrix is on the left and which is on the right?

  • When multiplying numbers "swapping the order doesn't change the result"

    • E.g. 5 × 4 = 4 × 5

  • This is not true for matrix multiplication

    • 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

How can I multiply more than two matrices together?

  • When multiplying numbers "it doesn't matter which order you group operations into"

    • E.g. to do 8 x 9 x 10, either (8 x 9) x 10 or 8 x (9 x 10) works 

  • This is also true for matrix multiplication

    • (AB)C ≡ A(BC)

  • To multiply three matrices together

    • it's fine to start by multiplying the first two together

      • then multiplying the answer with the third matrix

    • or to start by multiplying the second two together

      • then multiplying the answer with the first matrix

  • Just don't switch the order

    • A(BC) is not the same as (BC)A

What is special about the identity matrix?

  • Multiplying any 2×2 or 3×3 matrix by the corresponding identity matrix leaves it unchanged

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

  • For example, in the 2×2 case

    • 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 two matrices on the left side of each equation

  • The 2×2 identity matrix also leaves a 2×1 matrix unchanged

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

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 row 8 end table close parentheses, find the following:

(i) bold PR 

(ii) bold PQ 

(iii) bold Q squared

Answer:

(i)  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 row 8 end table close parentheses

Multiply the matrices

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

Simplify

Error converting from MathML to accessible text.

(ii)  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

Error converting from MathML to accessible text.

(iii)  Write out bold Q to the power of bold 2 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

Error converting from MathML to accessible text.

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.

Answer:

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

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 equals 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

equals open parentheses table row 4 0 row 0 4 end table close parentheses

Write in terms of the identity matrix,  bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses by factoring 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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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Jamie Wood

    Reviewer: Jamie Wood

    Expertise: Maths Content Creator

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.