Solving Matrix Equations (Edexcel IGCSE Maths B): Revision Note

Solving matrix equations with inverses

• Inverses can be used to rearrange and solve equations with matrices

• This relies on the essential properties of inverses and of the identity matrix:

  • ${\mathbf{AA}}^{-1}={\mathbf{A}}^{-1}\mathbf{A}=\mathbf{I}$

  • $\mathbf{AI}=\mathbf{IA}=\mathbf{A}$

• For example, to solve the matrix equation $\mathbf{AB}=\mathbf{C}$ for $\mathbf{B}$

  • First multiply both sides of the equation 'from the left' by ${\mathbf{A}}^{-1}$

    • ${\mathbf{A}}^{-1}\mathbf{AB}={\mathbf{A}}^{-1}\mathbf{C}$

  • But ${\mathbf{A}}^{-1}\mathbf{A}=\mathbf{I}$, so

    • $\mathbf{IB}={\mathbf{A}}^{-1}\mathbf{C}$

  • And $\mathbf{IB}=\mathbf{B}$, so

    • $\mathbf{B}={\mathbf{A}}^{-1}\mathbf{C}$

  • Then multiply together the two matrices on the right-hand side to find $\mathbf{B}$ explicitly

• Similarly, you can solve the matrix equation $\mathbf{BA}=\mathbf{C}$ for $\mathbf{B}$

  • Though here you will need to multiply 'from the right' by ${\mathbf{A}}^{-1}$

$\begin{array}{rcl}{\mathbf{BAA}}^{-1}& =& {\mathbf{CA}}^{-1}\\ \mathbf{BI}& =& {\mathbf{CA}}^{-1}\\ \mathbf{B}& =& {\mathbf{CA}}^{-1}\end{array}$

• This is similar to solving a regular equation by 'doing the same thing to both sides'

  • Except that order of multiplication matters with matrices

    • Multiplying 'from the left' is not the same as multiplying 'from the right'

  • For example, if you tried to solve $\mathbf{BA}=\mathbf{C}$ for $\mathbf{B}$ by multiplying 'from the left' by ${\mathbf{A}}^{-1}$

    • you would get ${\mathbf{A}}^{-1}\mathbf{BA}={\mathbf{A}}^{-1}\mathbf{C}$

    • which does not simplify any further