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IBMaths: AI HLDPRevision Notes1. Number & Algebra1.7 Matrices1.7.3 Determinants & Inverses

Determinants & Inverses (DP IB Maths: AI HL)

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Naomi C

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Naomi C

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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 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

  • 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
  • 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
  • When finding the determinant of a multiple of a matrix or the product of two matrices:
    • 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 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

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

How do I find the inverse of a matrix?

  • 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 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 comma space a d not equal to b c

  • 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
  • The inverse of a square matrix bold italic Ais the matrix bold italic A to the power of negative 1 end exponent such that 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
    • As a result of this property:
      • 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 (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 (post-multiplying by bold italic A to the power of negative 1 end exponent)

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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    Naomi C

    Author: Naomi C

    Expertise: Maths

    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.