A LevelFurther Maths: Core PureEdexcelRevision Notes2. Matrices2.1 Properties of Matrices2.1.3 Inverses of Matrices

Inverses of Matrices (Edexcel A Level Further Maths: Core Pure)

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Inverse of a Matrix

What is an 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 inverse of a square matrix bold italic Ais denoted as the matrix 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
  • You can use your calculator to find the inverse of matrices
    • You need to know how to find the inverse of 2x2 and 3x3 matrices by hand
  • Inverses can be used to rearrange equations with matrices:
    • 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)
  • The inverse of a product of matrices is the product of the inverse of the matrices in reverse order:
    • open parentheses bold italic A bold italic B close parentheses to the power of negative 1 end exponent equals bold italic B to the power of negative 1 end exponent bold italic A to the power of negative 1 end exponent

Examiner Tip

  • Many past exam questions exploit the property bold italic M bold italic M to the power of negative 1 end exponent equals bold italic I
    • these typically start with two, seemingly, unconnected matrices
      • M and N, say, possibly with some unknown elements
    • the result of MN is often a scalar multiple of I, kI say
    • so M and N are (almost) inverses of each other
      • You are expected to deduce bold italic M to the power of negative 1 end exponent equals 1 over k bold italic N
    • Look out for and practise this style of question, they are very common

Worked example

Consider the matrices bold italic M equals open parentheses table row p cell negative 2 end cell cell negative 3 end cell row cell negative 3 end cell 0 p row 1 cell 2 p end cell 1 end table close parentheses and bold italic N equals open parentheses table row cell negative 4 p end cell cell negative 10 end cell cell negative 2 p end cell row 5 5 5 row cell negative 12 end cell cell negative 5 p end cell cell negative 6 end cell end table close parentheses, where space p is a constant.

a)
Find bold italic M bold italic N, writing the elements in terms of space p where necessary.rn-2-1-properties-of-matrices-copy
b)
In the case space p equals 2, deduce the matrix bold italic M to the power of negative 1 end exponent. 

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Finding the Inverse of a 2x2 Matrix

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 leading diagonal
    • Change the signs of the other two entries
    • Divide by the determinant

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

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.

rn-1-7-matrices-copy

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

Finding the Inverse of a 3x3 Matrix

How do I find the inverse of a 3x3 matrix?

  • This is easiest to see with an example
    • Use the matrix bold italic A equals open parentheses table row 2 0 3 row cell negative 3 end cell 1 cell negative 1 end cell row 4 2 cell negative 2 end cell end table close parentheses
  • STEP 1
    Find the determinant of a 3x3 matrix
    • The inverse only exists if the determinant is non-zero
      • e.g.  det bold italic A equals 2 open vertical bar table row 1 cell negative 1 end cell row 2 cell negative 2 end cell end table close vertical bar minus 0 open vertical bar table row cell negative 3 end cell cell negative 1 end cell row 4 cell negative 2 end cell end table close vertical bar plus 3 open vertical bar table row cell negative 3 end cell 1 row 4 2 end table close vertical bar equals 2 left parenthesis 0 right parenthesis plus 3 left parenthesis negative 10 right parenthesis equals negative 30                
  • STEP 2
    Find the minor for every element in the matrix.
    • You will sometimes see this written as a huge matrix – like below
      This is called the matrix of minors and is often denoted by M
      With pen and paper, this can get quite large and cumbersome to work with so you may prefer to lay the minors out separately and form M at the end 
      • e.g. bold italic M equals open parentheses table row cell open vertical bar table row 1 cell negative 1 end cell row 2 cell negative 2 end cell end table close vertical bar end cell cell open vertical bar table row cell negative 3 end cell cell negative 1 end cell row 4 cell negative 2 end cell end table close vertical bar end cell cell open vertical bar table row cell negative 3 end cell 1 row 4 2 end table close vertical bar end cell row cell open vertical bar table row 0 3 row 2 cell negative 2 end cell end table close vertical bar end cell cell open vertical bar table row 2 3 row 4 cell negative 2 end cell end table close vertical bar end cell cell open vertical bar table row 2 0 row 2 2 end table close vertical bar end cell row cell open vertical bar table row 0 3 row 1 cell negative 1 end cell end table close vertical bar end cell cell open vertical bar table row 2 3 row cell negative 3 end cell cell negative 1 end cell end table close vertical bar end cell cell open vertical bar table row 2 0 row cell negative 3 end cell 1 end table close vertical bar end cell end table close parentheses equals open parentheses table row 0 10 cell negative 10 end cell row cell negative 6 end cell cell negative 16 end cell 4 row cell negative 3 end cell 7 2 end table close parentheses
  • STEP 3
    Find the matrix of cofactors, often denoted by C, by combining the matrix of signs, with the matrix of minors
    • The matrix of signs is open parentheses table row plus minus plus row minus plus minus row plus minus plus end table close parentheses 
      • e.g. bold italic C equals open parentheses table row cell plus left parenthesis 0 right parenthesis end cell cell negative left parenthesis 10 right parenthesis end cell cell plus left parenthesis negative 10 right parenthesis end cell row cell negative left parenthesis negative 6 right parenthesis end cell cell plus left parenthesis negative 16 right parenthesis end cell cell negative left parenthesis 4 right parenthesis end cell row cell plus left parenthesis negative 3 right parenthesis end cell cell negative left parenthesis 7 right parenthesis end cell cell plus left parenthesis 2 right parenthesis end cell end table close parentheses equals open parentheses table row 0 cell negative 10 end cell cell negative 10 end cell row 6 cell negative 16 end cell cell negative 4 end cell row cell negative 3 end cell cell negative 7 end cell 2 end table close parentheses
  • STEP 4
    Transpose the matrix of cofactors to form CT
    • This is sometimes called the adjugate of A
      • e.g. bold italic C to the power of straight T equals open parentheses table row 0 6 cell negative 3 end cell row cell negative 10 end cell cell negative 16 end cell cell negative 7 end cell row cell negative 10 end cell cell negative 4 end cell 2 end table close parentheses
  • STEP 5
    Find the inverse of A by dividing CT by the determinant of A
    • bold italic A to the power of negative 1 end exponent equals fraction numerator 1 over denominator det bold italic A end fraction bold italic C to the power of straight T
      • e.g. bold italic A to the power of negative 1 end exponent equals fraction numerator 1 over denominator negative 30 end fraction open parentheses table row 0 6 cell negative 3 end cell row cell negative 10 end cell cell negative 16 end cell cell negative 7 end cell row cell negative 10 end cell cell negative 4 end cell 2 end table close parentheses equals open parentheses table row 0 cell negative 2 over 15 end cell cell 1 over 10 end cell row cell 1 third end cell cell 8 over 15 end cell cell 7 over 30 end cell row cell 1 third end cell cell 2 over 15 end cell cell negative 1 over 15 end cell end table close parentheses
  • It is often convenient to leave A-1 as a (positive) scalar multiple of CT, rather than have a matrix full of fractions that can be awkward to read and follow
      • e.g.  bold italic A to the power of negative 1 end exponent equals 1 over 30 open parentheses table row 0 cell negative 6 end cell 3 row 10 16 7 row 10 4 cell negative 2 end cell end table close parentheses

Can I use my calculator to get the inverse of a matrix?

  • Yes, of course, but only where possible!
  • Questions with unknown elements will generally not be solvable directly on a calculator
    • If by the end of the questions, the unknowns have been found, you can then check your answers using the calculator
  • Some questions with purely numerical matrices may still ask you to show your full working without relying on calculator technology - but you can still use it at the end to check!
  • Two things to be very careful with when using your calculator
    • When entering values into a matrix, check and be clear as to where the cursor moves to after each element – does it move across or down?
    • When displaying a matrix many calculators will display values as (rounded/truncated) decimals; highlighting a particular one will show the value as an exact fraction

Examiner Tip

  • Do not worry too much about the various terms and language used in finding the inverse of a 3x3 matrix, learning and following the process (without a calculator) is more important
  • If a question says not to rely on "calculator technology" in your answer, you must show full working throughout
    • However, you can still use your calculator to check your work at the end
    • Consider the number of marks a question is worth for a clue as to how much working may be necessary

Worked example

Given that bold italic A equals open parentheses table row 2 cell negative 4 end cell 5 row 2 18 cell negative k end cell row 3 4 9 end table close parentheses, find bold italic A to the power of negative 1 end exponent in terms of k.

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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.