Determinants & Inverses (DP IB Applications & Interpretation (AI): HL): Revision Note

Determinants

What is a determinant?

• The determinant is a numerical value (positive or negative) calculated from the elements in a matrix and is used to find the inverse of a matrix

• You can only find the determinant of a square matrix

• The determinant of the square matrix $\mathit{A}$ is denoted by $\det \mathit{A}$ or $\left|\mathit{A}\right|$

• The method for finding the determinant of a $2\times 2$ matrix is given in your formula booklet: 

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

• For example, the determinant of $\left(\begin{array}{cc}3& -2\\ 5& 4\end{array}\right)$ is

  • $3\times 4-\left(-2\right)\times 5=22$

What are the properties of determinants?

• The determinant of an identity matrix is $\det \left(\mathit{I}\right)=1$

• The determinant of a zero matrix is $\det \left(\mathit{O}\right)=0$

• The determinant of a product of matrices is equal to the product of their determinants

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

• Multiplying a matrix by a scalar affects the determinant

  • $\det \left(k\mathit{A}\right)=k^2\det \left(\mathit{A}\right)$(for a $2\times 2$ matrix)

  • $\det \left(k\mathit{A}\right)=k^n\det \left(\mathit{A}\right)$(for a $n\times n$ matrix)

Inverse matrices

What is the inverse of a matrix?

• Only square matrices can have inverses

  • Not all square matrices have inverses

• 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 an invertible square matrix $\mathit{A}$is denoted as ${\mathit{A}}^{-1}$ 

  • the product of these matrices is an identity matrix

    • $\mathit{A}{\mathit{A}}^{-1}={\mathit{A}}^{-1}\mathit{A}=\mathit{I}$

• Inverse matrices can be used to rearrange matrix equations

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

    • this is the result after pre-multiplying by ${\mathit{A}}^{-1}$

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

    • this is the result after post-multiplying by ${\mathit{A}}^{-1}$

How do I find the inverse of a matrix?

• The method for finding the inverse of a $2\times 2$ matrix is given in your formula booklet:

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