Proving Matrix Relationships (Edexcel International AS Further Maths) : Revision Note
Proving Matrix Relationships
What is a matrix relationship?
A matrix relationship is an equation that connects different matrices
For example, if A, B and C are matrices then possible relationships are
AB = C
A + 2B = C
Remember that in general matrix multiplication is not commutative
AB ≠ BA
ABC ≠ ACB ≠ BAC ≠ …
CBC does not simplify to C2B
Since C(BC) ≠ C(CB)
What matrix relationships do I need to know?
If A and B are matrices and I is the identity matrix, you need to know the following:
AA-1 = I
A-1A = I
IA = AI = A
(AB)-1 = B-1A-1
The order reverses
How do I prove matrix relationships?
You need to carefully pre-multiply (on the left) or post-multiply (on the right) both sides by matrices or their inverses
To make B the subject of AB = C
A-1AB = A-1C (pre-multiply by A-1)
IB = A-1C (form the identity)
B = A-1C (simplify)
To make A the subject of AB = C
ABB-1 = CB-1 (post-multiply by B-1)
AI = CB-1 (form the identity)
A = CB-1 (simplify)
How do I prove that (AB)-1 = B-1A-1?
Start by multiplying AB by its inverse to form the identity
AB(AB)-1 = I
Then make (AB)-1 the subject
Pre-multiply by A-1 and simplify
A-1AB(AB)-1 = A-1I
IB(AB)-1=A-1
B(AB)-1=A-1
Then pre-multiply by B-1 and simplify
B-1B(AB)-1=B-1A-1
I(AB)-1=B-1A-1
(AB)-1=B-1A-1
Examiner Tips and Tricks
Learn the formula (AB)-1 =B-1A-1 as you are not given it in the Formulae Booklet.
Show lots of steps when doing matrix algebra.
Examiners want to see pre- or post-multiplying clearly.
Worked Example
Let ,
and
be three matrices such that
where
is the identity matrix.
Prove that .
You need to use matrix algebra to make the subject
One possible way is to remove from the left by pre-multiplying both sides by
The forms the identity matrix,
, on the left
The simplifies to just
on the right
The on the left simplifies to just
Now post-multiply both sides by
The on the left forms the identity matrix
The on the left simplifies to
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