Edexcel A Level Further Maths: Core Pure:复习笔记2.1.3 Inverses of Matrices

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 , then  is invertible
    • If , then  is singular and does not have an inverse
  • The inverse of a square matrix is denoted as the matrix
  • The product of these matrices is an identity matrix,
  • 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:
    • (pre-multiplying by )
    • (post-multiplying by)
  • The inverse of a product of matrices is the product of the inverse of the matrices in reverse order:

Exam Tip

  • Many past exam questions exploit the property
    • 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
    • Look out for and practise this style of question, they are very common

Worked Example

Consider the matrices and , where  is a constant.

a) Find , writing the elements in terms of  where necessary.rn-2-1-properties-of-matrices-copy
b) In the case , deduce the matrix rn-2-1-properties-of-matrices


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  matrix is:
    • Switch the two entries on leading diagonal
    • Change the signs of the other two entries
    • Divide by the determinant

Worked Example

Consider the matrices , where  is a constant.

a) Find .


b) Given that find the value of .


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
  • STEP 1
    Find the determinant of a 3x3 matrix

    • The inverse only exists if the determinant is non-zero
      • e.g. 
  • 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.
  • 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
      • e.g.
  • STEP 4
    Transpose the matrix of cofactors to form

    • This is sometimes called the adjugate of A
      • e.g.
  • STEP 5
    Find the inverse of A by dividing CT by the determinant of A

      • e.g.
  • 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.

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

Exam 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 , find  in terms of .