# 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
• 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.
b) In the case , deduce the matrix

### 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 .