Inverse Matrices (Edexcel International A Level Further Maths)
Revision Note
Author
Mark CurtisExpertise
Maths
Determinant of a 2x2 Matrix
What is the determinant?
The determinant is a numerical value (positive or negative) calculated from elements of a square matrix
It is used to find the inverse of a matrix
For a 2 × 2 matrix, , the determinant, or , is given by
What properties of determinants do I need to know?
If then is called a singular matrix
If then is called a non-singular matrix
The determinant of the identity matrix is 1
The determinant of the zero matrix is 0
You do not need to know the following rules, but they can be good checks in an exam:
Worked Example
Consider the matrix , where is a constant.
Given that , find the value of .
Use that if then
Work out the determinant algebraically
Set this expression equal to -3 and solve for
Inverse of a 2x2 Matrix
What is the inverse of a matrix?
The inverse of a square matrix is the matrix which, when multiplied together (in either order), gives the identity matrix
has the same dimensions (order) as
How do I find the inverse of a 2x2 matrix?
To find the inverse of a 2 × 2 matrix:
Switch the two entries on leading diagonal (top-left to bottom right)
Change the signs of the other two entries
Divide by the determinant
Note that you cannot divide by zero
If , then is not invertible ( does not exist)
is singular
If , then is invertible
How do I find the inverse of a product of matrices?
The inverse of a product of matrices is the product of the inverses of the matrices in reverse order
Exam Tip
There are two ways to check whether your inverse matrix is correct:
use a calculator (they can find inverses),
or calculate to see if you get the identity .
Worked Example
Let where .
(a) Find , giving your answer in terms of .
Use that where
First find
Now divide by , swap and , and change signs of and
You can also write this as
(b) Verify that , where is the identity matrix.
Verify means check that it is true
Substitute and into the left-hand side and multiply out the matrices
The is the same as
Since in the question, cancel the 's
The right-hand side is the 2 × 2 identity matrix
Even though is also true, this question only asks for the order shown
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