Determinants of Matrices (Edexcel IGCSE Maths B): Revision Note

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.

$\mathbf{A}$ and $\mathbf{B}$ are two matrices such that the determinant of $\mathbf{A}$ is equal to the determinant of $\mathbf{B}$.

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

Answer:

Use $\det \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=ad-bc$ to find the determinants of the two matrices

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

$\det \mathbf{B}=2\times 1-4\times 5=-18$

Set the two determinants equal and solve for $p$

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

$p=3^2=9$

$p=9$