GCSEFurther MathsAQARevision Notes5. Matrix TransformationsMatrix MultiplicationMultiplying Matrices

Multiplying Matrices (AQA GCSE Further Maths)

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Multiplying Matrices

How do I multiply a 2x2 matrix by a 2x1 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 2x2 matrix by another 2x2 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
  • The process becomes more natural the more times you do it!

How do I square a 2x2 matrix?

  • Do not square each individual element
  • 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 after squaring a matrix

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 is commutative
    • 4 ÷ 2 ≠ 2 ÷ 4 and 5 - 3 ≠ 3 - 5
      • Division and subtraction of numbers is not commutative
  • Matrix multiplication is not commutative
    • 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 x 9 x 10, either (8 x 9) x 10 or 8 x (9 x 10) works 
      • Multiplication and addition of numbers is associative
    • (8 ÷ 4) ÷ 2 ≠ 8 ÷ (4 ÷ 2) and (5 - 4) - 3 ≠ 5 - (4 - 3)
      • Division and subtraction of numbers is not associative
  • Matrix multiplication is associative
    • (AB)C ≡ A(BC)
  • To multiply three matrices together, it's fine to start by multiplying the first two together, or to start by multiplying the second two together
    • Just don't switch the order
      • A(BC) is not (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 row 8 end table close parentheses, find the following:

(i)
bold PR 
(ii)
bold PQ 
(iii)
bold Q squared
 
(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

begin bold style stretchy left parenthesis table row 38 row cell negative 20 end cell end table stretchy right parenthesis end style
(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

begin bold style stretchy left parenthesis table row 19 cell negative 13 end cell row cell negative 10 end cell 10 end table stretchy right parenthesis end style
(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

stretchy left parenthesis table row 5 cell negative 35 end cell row 28 cell negative 16 end cell end table stretchy right parenthesis

The Identity Matrix

What is the Identity Matrix?

  • The identity matrix, I, is a 2×2 matrix with 1s along the diagonal from top-left to bottom-right and zeros everywhere else
    • bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses
  • Multiplying any 2×2 matrix by the identity matrix leaves it unchanged
    • 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 two matrices on the left
  • The identity matrix is an important matrix which you should know (or recognise as 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

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

Forming Equations With Matrices

How do I form equations using matrices?

  • Matrices with letters as elements can lead to algebra
  • Multiply any matrices together first, then equate each element
    • elements on the left-hand side must equal their corresponding element on the right-hand side
    • if open parentheses table row a b row c d end table close parentheses equals open parentheses table row p q row r s end table close parentheses then a = p, b = q, c = r and d = s
  • Harder questions may lead to simultaneous equations being formed
  • Calculations may refer to I; the identity matrix
    • bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses

Worked example

If open parentheses table row a 3 row 9 b end table close parentheses open parentheses table row 2 1 row 3 0 end table close parentheses equals k bold I, find the values of a, b and k.

 

Multiply the matrices on the left-hand side, and write bold I as a matrix

open parentheses table row cell open parentheses a cross times 2 space plus space 3 cross times 3 close parentheses end cell cell open parentheses a cross times 1 space plus space 3 cross times 0 close parentheses end cell row cell open parentheses 9 cross times 2 space plus space b cross times 3 close parentheses end cell cell open parentheses 9 cross times 1 space plus space b cross times 0 close parentheses end cell end table close parentheses equals k open parentheses table row 1 0 row 0 1 end table close parentheses

Simplify, then multiply the right-hand side

open parentheses table row cell 2 a plus 9 end cell a row cell 18 plus 3 b end cell 9 end table close parentheses equals open parentheses table row k 0 row 0 k end table close parentheses

Equate the top-right elements

a equals 0

Equate the top-left elements (use a equals 0)

table row cell 2 a plus 9 end cell equals k row cell 2 open parentheses 0 close parentheses plus 9 end cell equals k row 9 equals k end table

Equate the bottom-left elements then solve

table row cell 18 plus 3 b end cell equals 0 row cell 3 b end cell equals cell negative 18 end cell row b equals cell negative 6 end cell end table

We have found all the unknowns, but we should also equate the bottom right-elements as a check

9 equals k

table row bold italic a bold equals bold 0 row bold italic b bold equals cell bold minus bold 6 end cell row bold italic k bold equals bold 9 end table

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    Mark

    Author: Mark

    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.