Edexcel A-Level Mathematics AS Fp1 5 Matrix Algebra Questions

Practise matrix operations with constants, including products, determinants, singularity tests and inverses of 2 by 2 matrices.

Exam points

multiply matrices with entries in k and simplify each entry of the product
use a 2 by 2 determinant to test singularity or solve for a constant
find a 2 by 2 inverse using the determinant and adjugate matrix

Covered syllabus points

FP1.5.1 - Addition and subtraction of matrices FP1.5.2 - Multiplication of a matrix by a scalar FP1.5.3 - Products of matrices FP1.5.4 - Evaluation of 2 × 2 determinants FP1.5.5 - Inverse of 2 × 2 matrices

[Maximum number: 1]

\mathbf{A}=\left(\begin{array}{cc} -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{array}\right)

Determine the matrix $\mathbf{A}^{2}$

$A^2= \left(\begin{smallmatrix} \frac12 & \frac{\sqrt3}{2}\\ -\frac{\sqrt3}{2} & \frac12 \end{smallmatrix}\right)$

[Maximum number: 5]

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

(a)

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

[ 3 ]

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

(b)

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

[ 2 ]

The determinant is $k^2+2k+4$ .

$\mathbf M^{-1} =\frac{1}{k^2+2k+4} \left(\text{row }1: k+4,\ -k;\ \text{row }2: -k-7,\ 2k+1\right).$

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Covered syllabus points

Question 1

Question 2

Question (a)

Question (b)