Edexcel A-Level Mathematics AS Fp1 5 4 Evaluation of 2 2 Determinants Questions

Practise Edexcel IAL FP1.5.4 by evaluating 2×2 determinants with ad−bc, setting them to zero for singularity and analysing determinant inequalities.

Exam points

evaluate a 2×2 determinant with ad−bc and simplify expressions involving parameters
set det M or det AB to zero to find parameter values that make a matrix singular
solve determinant inequalities to determine parameter ranges where det M is positive or non-zero

[Maximum number: 3]

$\mathbf M=\left(\text{row }1:2k+1,\ k;\ \text{row }2:k+7,\ k+4\right),$
where k is a constant.

Show that M is non-singular for all real values of k.

$\det\mathbf M=(2k+1)(k+4)-k(k+7)=k^2+2k+4.$
M1

$k^2+2k+4=(k+1)^2+3>0$
for all real k. M1

Therefore $\det\mathbf M\neq 0$ , so $\mathbf M$ is non-singular for all real k. A1

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Edexcel A-Level Mathematics AS Fp1 5 4 Evaluation of 2 2 Determinants Questions question 1