Proving Matrix Relationships (Edexcel International A Level Further Maths)

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