CAIE A-Level Further Mathematics 2.2.2 Determinants and Inverses

Practise using a 3 × 3 determinant to locate exceptional parameter values, then testing consistency and interpreting the resulting arrangement of planes.

Exam points

set the coefficient determinant to zero to find parameters for which no unique solution exists
substitute each exceptional value and reduce the equations far enough to prove consistency or contradiction
translate the algebra into the correct plane geometry, such as a common line or parallel planes

[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)

$\begin{aligned} x =\frac{11}{2}+\frac{8}{a} \left|\begin{array}{ccc}a 2 a+5 a+1 0 -4 0 3 -1\end{array}\right|=4 a \neq 0 \text { or } \quad y =-\frac{1}{2} z =-\frac{9}{2}\end{aligned}$

M1 A1

Shows that determinant is non-zero or
solves equations and finds at least two
correct values.

The three planes intersect at a single point.

B1

3

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2.2.2—Determinants and inverses question 1