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
- If
- 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:
Examiner 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
- You are expected to deduce
- Look out for and practise this style of question, they are very common
- these typically start with two, seemingly, unconnected matrices
Worked example
Consider the matrices and
, where
is a constant.
a)
Find
, writing the elements in terms of
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b)
In the case
, deduce the matrix
.