IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraMatricesDeterminants & Inverses

Determinants & Inverses (DP IB Applications & Interpretation (AI)): Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Determinants

What is a determinant?

  • The determinant is a numerical value (positive or negative) calculated from the elements in a matrix and is used to find the inverse of a matrix

  • You can only find the determinant of a square matrix

  • The determinant of the square matrix bold italic A is denoted by det bold space bold italic A or open vertical bar bold italic A close vertical bar

  • The method for finding the determinant of a 2 cross times 2 matrix is given in your formula booklet: 

bold italic A equals open parentheses table row a b row c d end table close parentheses rightwards double arrow det space bold italic A equals open vertical bar bold italic A close vertical bar equals a d minus b c

  • For example, the determinant of open parentheses table row 3 cell negative 2 end cell row 5 4 end table close parentheses is

    • 3 cross times 4 minus open parentheses negative 2 close parentheses cross times 5 equals 22

Examiner Tips and Tricks

You only need to be able to find the determinant of a 2 cross times 2 matrix by hand. For larger n cross times n matrices you are expected to use your GDC.

What are the properties of determinants?

  • The determinant of an identity matrix is det space left parenthesis bold italic I right parenthesis equals 1

  • The determinant of a zero matrix is det space left parenthesis bold italic O right parenthesis equals 0

  • The determinant of a product of matrices is equal to the product of their determinants

    • det space left parenthesis bold italic A bold italic B right parenthesis equals det space left parenthesis bold italic A right parenthesis cross times det space left parenthesis bold italic B right parenthesis

  • Multiplying a matrix by a scalar affects the determinant

    • det space left parenthesis k bold italic A right parenthesis equals k squared space det space left parenthesis bold italic A right parenthesis(for a 2 cross times 2 matrix)

    • det space left parenthesis k bold italic A right parenthesis equals k to the power of n space det space left parenthesis bold italic A right parenthesis(for a n cross times n matrix)

Worked Example

Consider the matrix bold italic A equals open parentheses table row 3 cell negative 6 end cell row p 7 end table close parentheses, where p element of straight real numbers is a constant.

a) Given that det space bold italic A equals negative 3, find the value of p.

1-7-3-ib-ai-hl-determinants--inverses-we-1a

b) Find the determinant of 4 bold italic A.

1-7-3-ib-ai-hl-determinants--inverses-we-1b

Inverse matrices

What is the inverse of a matrix?

  • Only square matrices can have inverses

    • Not all square matrices have inverses

  • The determinant can be used to find out if a matrix is invertible or not:

    • If det space bold italic A not equal to 0, then bold italic A is invertible

    • If det space bold italic A equals 0, then bold italic A is singular and does not have an inverse

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

    • the product of these matrices is an identity matrix

      • bold italic A bold italic A to the power of negative 1 end exponent equals bold italic A to the power of negative 1 end exponent bold italic A equals bold italic I

  • Inverse matrices can be used to rearrange matrix equations

    • bold italic A bold italic B equals bold italic C rightwards double arrow bold italic B equals bold italic A to the power of negative 1 end exponent bold italic C 

      • this is the result after pre-multiplying by bold italic A to the power of negative 1 end exponent

    • bold italic B bold italic A equals bold italic C rightwards double arrow bold italic B equals bold italic C bold italic A to the power of negative 1 end exponent

      • this is the result after post-multiplying by bold italic A to the power of negative 1 end exponent

How do I find the inverse of a matrix?

  • The method for finding the inverse of a 2 cross times 2 matrix is given in your formula booklet:

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

You only need to be able to find the inverse of a 2 cross times 2 matrix by hand. For larger n cross times n matrices you are expected to use your GDC.

Worked Example

Consider the matrices bold italic P equals open parentheses table row 4 cell negative 2 end cell row 8 2 end table close parentheses, bold italic Q equals open parentheses table row k 6 row cell negative 5 end cell 3 end table close parentheses and bold italic R equals open parentheses table row 18 18 row 6 54 end table close parentheses, where k is a constant.

a) Find bold italic P to the power of negative 1 end exponent.

1-7-3-ib-ai-hl-determinants--inverses-we-2a

b) Given that bold italic P bold italic Q equals bold italic R find the value of k.

1-7-3-ib-ai-hl-determinants--inverses-we-2b

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.