Edexcel IGCSE Math B 5.E Determinants & inverses Question Bank

Practise determinants and inverse matrices, including 2x2 inverses, matrix equations, transformations, and determinant conditions.

Exam points

find a 2x2 inverse by calculating the determinant and writing the adjugate matrix
use A=BC or similar equations to isolate an unknown matrix with an inverse
link determinants to transformations or matrix products to solve for constants

[Maximum number: 3]

$\mathbf A=\begin{pmatrix}2k^2&k-9\\-3k&k+1\end{pmatrix}, \qquad \mathbf B=\begin{pmatrix}1&-5\\3&k\end{pmatrix}.$
The determinant of matrix $\mathbf A$ is equal to the determinant of matrix $\mathbf B$ .

Show that k is a root of the equation
$2k^3+5k^2-28k-15=0.$

$\det A=2k^{2}(k+1)-(-3k)(k-9) = 2k^{3}+5k^{2}-27k.$ $\det B=k-3(-5)=k+15.$
Equating determinants:
$2k^{3}+5k^{2}-27k=k+15.$
Hence
$2k^{3}+5k^{2}-28k-15=0.$
Required result shown.

Marking notes:
The full equation including =0 must be seen.

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5.E Determinants and inverses question 1