Pearson Edexcel IAL Mathematics FP3.6.5 Inverse of 3 × 3 matrices

Practise finding inverses of 3 × 3 matrices using determinants, cofactors and adjugates, then applying them in transformations.

Exam points

find M^-1 from the determinant, cofactor matrix and adjugate for a 3 × 3 matrix
use an inverse matrix to link transformed lines, tetrahedron volumes or parameters
apply (AB)^-1 = B^-1A^-1 when reversing a combined transformation

[Maximum number: 4]

The matrix M is given by

\left(\begin{array}{rrr} 2 & 0 & -1 \\ k & 3 & 2 \\ -2 & 1 & k \end{array}\right)

find $\mathbf{M}^{-1}$ in terms of k.

The points O(0,0,0), A(4,-8,3), B(-2,5,-4) and C(4,-6,8) are the vertices of a tetrahedron T.

The transformation represented by matrix M transforms T to a tetrahedron with volume 50

$\operatorname{adj}M= \left(\begin{smallmatrix} 3k-2&-1&3\\ -k^2-4&2k-2&-4-k\\ k+6&-2&6 \end{smallmatrix}\right)$ $M^{-1}=\frac{1}{5k-10} \left(\begin{smallmatrix} 3k-2&-1&3\\ -k^2-4&2k-2&-4-k\\ k+6&-2&6 \end{smallmatrix}\right)$

M1 M1 dM1 A1

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FP3.6.5 - Inverse of 3 × 3 matrices question 1