Edexcel A-Level Mathematics A2 Fp3 6 Further Matrix Algebra Questions

Practise further matrix algebra with 3×3 determinants, inverses, transformations, eigenvectors and diagonalisation of symmetric matrices.

Exam points

evaluate det M to decide when a 3×3 matrix has no inverse or changes volume scale
find M^-1 using cofactors or an adjugate, including entries depending on a parameter
use eigenvalues and normalised eigenvectors to build D and an orthogonal matrix P

Covered syllabus points

FP3.6.1 - Linear transformations of column FP3.6.2 - Combination of transformations FP3.6.3 - Transpose of a matrix FP3.6.4 - Evaluation of 3 × 3 determinants FP3.6.5 - Inverse of 3 × 3 matrices FP3.6.6 - Inverse transformations and combinations FP3.6.7 - Eigenvalues and eigenvectors FP3.6.8 - Reduction of symmetric matrices

[Maximum number: 4]

\mathbf{T}=\left(\begin{array}{lll} 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array}\right) \quad \mathbf{U}=\left(\begin{array}{rrr} 6 & -1 & -4 \\ 15 & c & -9 \\ -8 & a & 5 \end{array}\right)

where a, b and c are constants.
Given that T U=I

determine the value of a, the value of b and the value of c

\mathbf{TU}=\mathbf I

6a+60-8b=0,\quad -4a-36+5b=1

6a-8b=-60,\quad -4a+5b=37

a=2,\quad b=9

-2+3c+7a=0

c=-4

M1: obtains at least two equations with at least one correct.\ndM1: obtains values for two of a,b,c.\nA1 A1: a=2,b=9,c=-4.

[Maximum number: 11]

The matrix M is given by

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

(a)

Show that $\operatorname{det} \mathbf{M}=5 k-10$

Given that $k \neq 2$

[ 2 ]

$\det M= \begin{vmatrix} 2&0&-1\\ k&3&2\\ -2&1&k \end{vmatrix} =2\begin{vmatrix}3&2\\1&k\end{vmatrix} -0\begin{vmatrix}k&2\\-2&k\end{vmatrix} (-1)\begin{vmatrix}k&3\\-2&1\end{vmatrix}$ =2(3k-2)-(k+6)=6k-4-k-6=5k-10

M1 A1*

(b)

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

[ 4 ]

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

(c)

Determine the possible values of k.

[ 5 ]

$\text{Volume of }T=\frac16|\mathbf a\cdot(\mathbf b\times\mathbf c)| =\frac{20}{3}$ $\text{Volume image}=|\det M|\cdot\frac{20}{3} \Rightarrow \frac{20}{3}|5k-10|=50$ $k=\frac72\quad\text{or}\quad k=\frac12$

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)

Question (c)