CAIE A-Level Further Math A2 2.2.2 Determinants and Inverses Questions

Practise calculating determinants, using inverse matrices to solve equations and interpreting singular systems with Further Mathematics Paper 2 questions.

Exam points

Calculate determinants and use inverse matrices to solve equations and interpret singular systems.

[Maximum number: 3]

It is given that a is a positive constant.

Show that the system of equations

\begin{aligned} a x+(2 a+5) y+(a+1) z & =1, \\ -4 y & =2, \\ 3 y-z & =3, \end{aligned}

has a unique solution and interpret this situation geometrically.

The matrix A is given by

\mathbf{A}=\left(\begin{array}{ccc} a & 2 a+5 & a+1 \\ 0 & -4 & 0 \\ 0 & 3 & -1 \end{array}\right)

$\begin{aligned} x =\frac{11}{2}+\frac{8}{a} \left|\begin{array}{ccc}a 2 a+5 a+1 0 -4 0 3 -1\end{array}\right|=4 a \neq 0 \text { or } \quad y =-\frac{1}{2} z =-\frac{9}{2}\end{aligned}$
Shows that determinant is non-zero or
solves equations and finds at least two
correct values.

The three planes intersect at a single point.
3

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CAIE A-Level Further Math A2 2.2.2 Determinants and Inverses Questions question 1