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

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:
  • 

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

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