International A LevelFurther MathsEdexcelFurther Pure 1Revision NotesMatricesMatrix AlgebraIntroduction to Matrices

Introduction to Matrices (Edexcel International A Level Further Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Introduction to Matrices

What is a matrix?

  • A matrix is a rectangular grid (array) of elements (numbers or letters) arranged in rows and columns

    • The plural of matrix is matrices

  • The order (dimensions) of a matrix is its number of rows × number of columns

    • a 2 × 1 matrix is open parentheses table row a row b end table close parentheses

      • this is also called a column matrix or a column vector

    • a 2 × 2 matrix is open parentheses table row a b row c d end table close parentheses

      • this is called a square matrix

  • A bold capital letter is often used to represent a matrix

    •  bold A equals open parentheses table row 5 2 row 0 4 end table close parentheses, bold B equals open parentheses table row 4 row 3 end table close parentheses

  • 2D coordinates can be written as a column matrix

    • The point open parentheses 3 comma space 5 close parentheses is open parentheses table row 3 row 5 end table close parentheses 

  • You can use subscript notation to refer to elements in a matrix

    • Matrix bold A is bold A equals left parenthesis a subscript i comma j end subscript right parenthesis where i equals 1 comma space 2 comma space 3 comma space... comma space m and j equals 1 comma space 2 comma space 3 comma space... comma space n

      • a subscript i comma j end subscript refers to the element in row i, column j

      • The order of bold A is m cross times n (rows × columns)

A matrix written in subscript notation

What type of matrices do I need to know?

  • A column matrix (or column vector) is a matrix with a single column

    • Order m cross times 1

  • A row matrix is a matrix with a single row

    • Order 1 cross times n

  • A square matrix is one in which the number of rows is equal to the number of columns

    • Order n cross times n

  • Two matrices are equal when they are of the same order and their corresponding elements are equal

    • a subscript i comma j end subscript equals b subscript i comma j end subscript for all elements

  • A zero matrix, bold 0, is a matrix in which all the elements are zero

    • For example, the 2 × 2 zero matrix is bold 0 equals open parentheses table row 0 0 row 0 0 end table close parentheses

  • An identity matrix, bold I, is a square matrix in which all elements along the leading diagonal (top-left to bottom right) are 1

    • The rest of the elements are zero

    • For example, the 2 × 2 identity matrix is  bold I equals open parentheses table row 1 0 row 0 1 end table close parentheses

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

Basic Operations with Matrices

How do I multiply a matrix by a scalar?

  • To multiply any matrix by a scalar (a number), multiply each element by that scalar 

    • If bold A equals open parentheses table row 5 2 row 0 4 end table close parentheses then 2 bold A equals 2 open parentheses table row 5 2 row 0 4 end table close parentheses equals open parentheses table row cell 2 cross times 5 end cell cell 2 cross times 2 end cell row cell 2 cross times 0 end cell cell 2 cross times 4 end cell end table close parentheses equals open parentheses table row 10 4 row 0 8 end table close parentheses

  • Multiplying by a negative scalar changes the sign of each element in the matrix

  • Lower case letters often refer to scalar multiples

    • k bold A is the matrix bold A multiplied by the scalar k

How do I add and subtract matrices?

  • Two matrices of the same order can be added (or subtracted) by adding (or subtracting) corresponding elements

    • The answer is a matrix of the same order

    • For example, open parentheses table row 1 2 row cell negative 3 end cell 0 row 2 1 end table close parentheses plus open parentheses table row 3 0 row 3 cell negative 1 end cell row 4 1 end table close parentheses equals open parentheses table row cell 1 plus 3 end cell cell 2 plus 0 end cell row cell negative 3 plus 3 end cell cell 0 minus 1 end cell row cell 2 plus 4 end cell cell 1 plus 1 end cell end table close parentheses equals open parentheses table row 4 2 row 0 cell negative 1 end cell row 6 2 end table close parentheses

What properties of matrix addition do I need to know?

  • bold A plus bold B equals bold B plus bold A

    • Matrix addition is commutative

      • You can swap the order

    • Matrix subtraction is not commutative

      • bold A minus bold B not equal to bold B minus bold A

  • bold A minus bold B equals bold A plus left parenthesis negative bold B right parenthesis

    • Subtraction is the same as adding a negative

  • bold A plus left parenthesis bold B plus bold C right parenthesis equals left parenthesis bold A plus bold B right parenthesis plus bold C

    • Matrix addition is associative

      • To add three matrices, you can start with the first two, or the last two

    • Matrix subtraction is not associative

      • bold A minus left parenthesis bold B minus bold C right parenthesis not equal to left parenthesis bold A minus bold B right parenthesis minus bold C

      • Try expanding the brackets to see

  • bold A plus bold 0 equals bold A

    • Adding the zero matrix has no effect

  • bold 0 minus bold A equals negative bold A

Worked Example

Consider the matrices bold A equals open parentheses table row cell negative 4 end cell 2 row 7 3 row 1 cell negative 5 end cell end table close parentheses, bold B equals open parentheses table row 2 6 row 5 cell negative 9 end cell row cell negative 2 end cell cell negative 3 end cell end table close parentheses.

(a) Find bold A plus bold B.

The matrices have the same order (dimensions)
Corresponding elements can be added

bold A plus bold B equals open parentheses table row cell negative 4 plus 2 end cell cell 2 plus 6 end cell row cell 7 plus 5 end cell cell 3 plus open parentheses negative 9 close parentheses end cell row cell 1 plus open parentheses negative 2 close parentheses end cell cell negative 5 plus open parentheses negative 3 close parentheses end cell end table close parentheses

bold A plus bold B equals open parentheses table row cell negative 2 end cell 8 row 12 cell negative 6 end cell row cell negative 1 end cell cell negative 8 end cell end table close parentheses

(b) Find negative 10 bold B.

Multiply each element by -10

table row cell negative 10 bold B end cell equals cell negative 10 open parentheses table row 2 6 row 5 cell negative 9 end cell row cell negative 2 end cell cell negative 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 10 cross times 2 end cell cell negative 10 cross times 6 end cell row cell negative 10 cross times 5 end cell cell negative 10 cross times open parentheses negative 9 close parentheses end cell row cell negative 10 cross times open parentheses negative 2 close parentheses end cell cell negative 10 cross times open parentheses negative 3 close parentheses end cell end table close parentheses end cell end table

table row cell negative 10 bold B end cell equals cell open parentheses table row cell negative 20 end cell cell negative 60 end cell row cell negative 50 end cell 90 row 20 30 end table close parentheses end cell end table

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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.