IGCSEMathsEdexcelMaths BRevision NotesMatricesProperties of MatricesInverses of Matrices

Inverses of Matrices (Edexcel IGCSE Maths B): Revision Note

Exam code: 4MB1

Roger B

Written by: Roger B

Reviewed by: Jamie Wood

Updated on

Inverse of a 2x2 matrix

What is the inverse of a matrix?

  • Only a square matrix has an inverse

  • The inverse of a square matrix bold Ais denoted as the matrix bold A to the power of negative 1 end exponent 

    • The product of these matrices is the identity matrix, bold AA to the power of negative 1 end exponent equals bold A to the power of negative 1 end exponent bold A equals bold I

How do I find the inverse of a 2x2 matrix?

  • The method for finding the inverse of a 2 cross times 2 matrix is:

    • Switch the two entries on the top left-bottom right diagonal

    • Change the signs of the other two entries

    • Divide by the determinant a d minus b c

bold A equals open parentheses table row a b row c d end table close parentheses space space rightwards double arrow space space bold A to the power of bold minus bold 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses

Examiner Tips and Tricks

The formula for the inverse is on the course Formulae sheet, and will be provided for you in Paper 2 questions where it is needed.

Determinant questions can also appear in Paper 1, however. While these can usually be answered without having to use the inverse formula, it is still a good idea to know the formula going into the exam.

  • Note that if the determinant, a d minus b c, is equal to zero, then the term fraction numerator 1 over denominator a d minus b c end fraction in the formula is not defined

    • If this is the case, then the matrix does not have an inverse

    • You will not need to deal with this situation on the exam

Worked Example

bold A equals open parentheses table row 1 3 row 2 7 end table close parentheses space space space space space space space bold B equals open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses

Show that bold B is the inverse of bold A.

Answer:

Method 1

If you know the inverse formula, you can use it to calculate the inverse of bold A directly

  • The inverse of space open parentheses table row a b row c d end table close parentheses space is space fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses

table row cell bold A to the power of bold minus bold 1 end exponent end cell equals cell fraction numerator 1 over denominator 1 cross times 7 minus 3 cross times 2 end fraction open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 7 minus 6 end fraction open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell row blank equals cell 1 open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell row blank equals bold B end table

bold B is the inverse of bold A
 

Method 2

If you don't know the inverse formula, you can use the definition of the inverse

  • I.e. if bold A to the power of negative 1 end exponent is the inverse of bold A, then bold AA to the power of negative 1 end exponent equals bold A to the power of negative 1 end exponent bold A equals bold I

So you can show that bold B is the inverse of bold A by showing

  • either that bold AB equals bold I

  • or that bold BA equals bold I

table row bold AB equals cell open parentheses table row 1 3 row 2 7 end table close parentheses open parentheses table row 7 cell negative 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 cross times 7 plus 3 cross times open parentheses negative 2 close parentheses end cell cell 1 cross times open parentheses negative 3 close parentheses plus 3 cross times 1 end cell row cell 2 cross times 7 plus 7 cross times open parentheses negative 2 close parentheses end cell cell 2 cross times open parentheses negative 3 close parentheses plus 7 cross times 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 7 minus 6 end cell cell negative 3 plus 3 end cell row cell 14 minus 14 end cell cell negative 6 plus 7 end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell row blank equals bold I end table

bold AB is equal to the identity matrix, so bold B is the inverse of bold A

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Previous:Determinants of MatricesNext:Solving Matrix Equations
    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.