Inverses of Matrices (Edexcel A Level Further Maths: Core Pure): Revision Note

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 $\det \mathit{A}\ne 0$, then $\mathit{A}$ is invertible

  • If $\det \mathit{A}=0$, then $\mathit{A}$ is singular and does not have an inverse

• The inverse of a square matrix $\mathit{A}$is denoted as the matrix ${\mathit{A}}^{-1}$ 

• The product of these matrices is an identity matrix, $\mathit{A}{\mathit{A}}^{-1}={\mathit{A}}^{-1}\mathit{A}=\mathit{I}$

• 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:

  • $\mathit{A}\mathit{B}=\mathit{C}\Rightarrow \mathit{B}={\mathit{A}}^{-1}\mathit{C}$ (pre-multiplying by ${\mathit{A}}^{-1}$)

  • $\mathit{B}\mathit{A}=\mathit{C}\Rightarrow \mathit{B}=\mathit{C}{\mathit{A}}^{-1}$ (post-multiplying by ${\mathit{A}}^{-1}$)

• The inverse of a product of matrices is the product of the inverse of the matrices in reverse order:

  • ${\left(\mathit{A}\mathit{B}\right)}^{-1}={\mathit{B}}^{-1}{\mathit{A}}^{-1}$

Finding the inverse of a 2x2 matrix

How do I find the inverse of a 2x2 matrix?

• The method for finding the inverse of a $2\times 2$ matrix is:

  • Switch the two entries on leading diagonal

  • Change the signs of the other two entries

  • Divide by the determinant

$\mathit{A}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\Rightarrow {\mathit{A}}^{\mathbf{-}\mathbf{1}}=\frac{1}{\det \mathit{A}}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right),ad\ne bc$

Finding the inverse of a 3x3 matrix

How do I find the inverse of a 3x3 matrix?

• This is easiest to see with an example

  • Use the matrix $\mathit{A}=\left(\begin{array}{ccc}2& 0& 3\\ -3& 1& -1\\ 4& 2& -2\end{array}\right)$

• STEP 1 Find the determinant of a 3x3 matrix

  • The inverse only exists if the determinant is non-zero

    • e.g.  $\det \mathit{A}=2\left|\begin{array}{cc}1& -1\\ 2& -2\end{array}\right|-0\left|\begin{array}{cc}-3& -1\\ 4& -2\end{array}\right|+3\left|\begin{array}{cc}-3& 1\\ 4& 2\end{array}\right|=2\left(0\right)+3(-10)=-30$                

• STEP 2 Find the minor for every element in the matrix.

  • You will sometimes see this written as a huge matrix – like below This is called the matrix of minors and is often denoted by M With pen and paper, this can get quite large and cumbersome to work with so you may prefer to lay the minors out separately and form M at the end 

    • e.g. $\mathit{M}=\left(\begin{array}{ccc}\left|\begin{array}{cc}1& -1\\ 2& -2\end{array}\right|& \left|\begin{array}{cc}-3& -1\\ 4& -2\end{array}\right|& \left|\begin{array}{cc}-3& 1\\ 4& 2\end{array}\right|\\ \left|\begin{array}{cc}0& 3\\ 2& -2\end{array}\right|& \left|\begin{array}{cc}2& 3\\ 4& -2\end{array}\right|& \left|\begin{array}{cc}2& 0\\ 4& 2\end{array}\right|\\ \left|\begin{array}{cc}0& 3\\ 1& -1\end{array}\right|& \left|\begin{array}{cc}2& 3\\ -3& -1\end{array}\right|& \left|\begin{array}{cc}2& 0\\ -3& 1\end{array}\right|\end{array}\right)=\left(\begin{array}{ccc}0& 10& -10\\ -6& -16& 4\\ -3& 7& 2\end{array}\right)$

• STEP 3 Find the matrix of cofactors, often denoted by C, by combining the matrix of signs, with the matrix of minors

  • The matrix of signs is $\left(\begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}\right)$ 

    • e.g. $\mathit{C}=\left(\begin{array}{ccc}+\left(0\right)& -\left(10\right)& +(-10)\\ -(-6)& +(-16)& -\left(4\right)\\ +(-3)& -\left(7\right)& +\left(2\right)\end{array}\right)=\left(\begin{array}{ccc}0& -10& -10\\ 6& -16& -4\\ -3& -7& 2\end{array}\right)$

• STEP 4 Transpose the matrix of cofactors to form C^T

  • This is sometimes called the adjugate of A

    • e.g. ${\mathit{C}}^{\mathrm{T}}=\left(\begin{array}{ccc}0& 6& -3\\ -10& -16& -7\\ -10& -4& 2\end{array}\right)$

• STEP 5 Find the inverse of A by dividing C^T by the determinant of A

  • ${\mathit{A}}^{-1}=\frac{1}{\det \mathit{A}}{\mathit{C}}^{\mathrm{T}}$

    • e.g. ${\mathit{A}}^{-1}=\frac{1}{-30}\left(\begin{array}{ccc}0& 6& -3\\ -10& -16& -7\\ -10& -4& 2\end{array}\right)=\left(\begin{array}{ccc}0& -\frac{1}{5}& \frac{1}{10}\\ \frac{1}{3}& \frac{8}{15}& \frac{7}{30}\\ \frac{1}{3}& \frac{2}{15}& -\frac{1}{15}\end{array}\right)$

• It is often convenient to leave A^-1 as a (positive) scalar multiple of C^T, rather than have a matrix full of fractions that can be awkward to read and follow

  • e.g.  ${\mathit{A}}^{-1}=\frac{1}{30}\left(\begin{array}{ccc}0& -6& 3\\ 10& 16& 7\\ 10& 4& -2\end{array}\right)$

Can I use my calculator to get the inverse of a matrix?

• Yes, of course, but only where possible!

• Questions with unknown elements will generally not be solvable directly on a calculator

  • If by the end of the questions, the unknowns have been found, you can then check your answers using the calculator

• Some questions with purely numerical matrices may still ask you to show your full working without relying on calculator technology - but you can still use it at the end to check!

• Two things to be very careful with when using your calculator

  • When entering values into a matrix, check and be clear as to where the cursor moves to after each element – does it move across or down?

  • When displaying a matrix many calculators will display values as (rounded/truncated) decimals; highlighting a particular one will show the value as an exact fraction