Determinants of Matrices (Edexcel IGCSE Maths)

Determinant of a 2x2 Matrix

Spec point — 5E

What is a determinant?

• The determinant is a numerical value (positive or negative) calculated from the elements in a matrix

  • It is used to find the inverse of a matrix

• You can only find the determinant of a square matrix*

• For a $2\times2$ matrix $( \begin{gathered} a b \\ c d \end{gathered} )$

  • The determinant is equal to $ad-bc$

Exam Hint The formula for the determinant is on the course Formulae sheet, and will be provided for you in Paper 2 questions where it is needed.

However, determinant questions can also appear in Paper 1, so be sure you know the formula going into the exam.

• The notation $\mathrm{det}$ is sometimes used to indicate the determinant

  • So $\mathrm{det}A$ refers to the determinant of matrix $A$

  • E.g. if $A = ( \begin{gathered} 3 4 \\ - 2 & - 1 \end{gathered} )$

    • then $\det A = 3 \times ( - 1 ) - 4 \times ( - 2 ) = - 3 - ( - 8 ) = 5$

Worked Example $A$ and $B$ are two matrices such that the determinant of $A$ is equal to the determinant of $B$.

Given that $A = ( \begin{gathered} \sqrt{p} & - 1 \\ 3 - 7 \end{gathered} )$ and $B = ( \begin{gathered} 2 4 \\ 5 1 \end{gathered} )$, where $p$ is a positive integer, find the value of $p$.

Answer:

Use $\det ( \begin{gathered} a b \\ c d \end{gathered} ) = a d - b c$ to find the determinants of the two matrices

$\det A = \sqrt{p} \times ( - 7 ) - ( - 1 ) \times 3 = 3 - 7 \sqrt{p}$

$\mathrm{det}B=2\times1-4\times5=-18$

Set the two determinants equal and solve for $p$

$\begin{gathered} - 18 & = 3 - 7 \sqrt{p} \\ 7 \sqrt{p} & = 3 + 18 \\ 7 \sqrt{p} & = 21 \\ \sqrt{p} & = 3 \end{gathered}$

$p=3^{2}=9$

$p=9$