Inverse Matrices (Edexcel International AS Further Maths: Further Pure 1): Revision Note

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, $\mathbf{A}$, the determinant, $\det \left(\mathbf{A}\right)$ or $\left|\mathbf{A}\right|$, is given by

$\mathbf{A}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\Rightarrow \det \mathbf{A}=\left|\mathbf{A}\right|=ad-bc$

What properties of determinants do I need to know?

• If $\det \left(\mathbf{A}\right)=0$ then $\mathbf{A}$ is called a singular matrix

• If $\det \left(\mathbf{A}\right)\ne 0$ then $\mathbf{A}$ is called a non-singular matrix

• The determinant of the identity matrix is 1

  • $\det \left(\mathbf{I}\right)=1$

• The determinant of the zero matrix is 0

  • $\det \left(\mathbf{0}\right)=0$

• You do not need to know the following rules, but they can be good checks in an exam:

  • $\det \left(\mathbf{AB}\right)=\det \left(\mathbf{A}\right)\times \det \left(\mathbf{B}\right)$

  • $\det \left({\mathbf{A}}^{-1}\right)=\frac{1}{\det \left(\mathbf{A}\right)}$

Inverse of a 2x2 Matrix

What is the inverse of a matrix?

• The inverse of a square matrix $\mathbf{A}$is the matrix ${\mathbf{A}}^{-1}$ which, when multiplied together (in either order), gives the identity matrix

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

  • ${\mathbf{A}}^{-1}$  has the same dimensions (order) as $\mathbf{A}$

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

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

• Note that you cannot divide by zero

  • If $\det \mathbf{A}=0$, then $\mathbf{A}$ is not invertible ( ${\mathbf{A}}^{-1}$ does not exist)

    • $\mathbf{A}$ is singular

  • If $\det \mathbf{A}\ne 0$, then $\mathbf{A}$ 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

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