Edexcel A Level Further Maths
Topic questions Pure mathematics Inverse matrices
10 questions · 90 marks · about 2 hours
Determine the values of a for which the matrix M is singular.
Given that matrix M is non-singular and that
• det(M) = −108
• a 
0
determine the matrix M−1
Show that M is non-singular for all values of a.
Determine, in terms of a, M–1
Show that M is non-singular for all values of a.
Determine, in terms of a, M–1
In this question you must show all stages of your working.
A college offers only three courses: Construction, Design and Hospitality.
Each student enrols on just one of these courses.
In 2019, there was a total of 1110 students at this college.
There were 370 more students enrolled on Construction than Hospitality.
In 2020 the number of students enrolled on
• Construction increased by 1.25%
• Design increased by 2.5%
• Hospitality decreased by 2%
In 2020, the total number of students at the college increased by 0.27% to 2 significant figures.
Define, for each course, a variable for the number of students enrolled on that course in 2019.
Using your variables from part (a)(i), write down three equations that model this situation.
By forming and solving a matrix equation, determine how many students were enrolled on each of the three courses in 2019.
In this question you must show all stages of your working.
A college offers only three courses: Construction, Design and Hospitality.
Each student enrols on just one of these courses.
In 2019, there was a total of 1110 students at this college.
There were 370 more students enrolled on Construction than Hospitality.
In 2020 the number of students enrolled on
• Construction increased by 1.25%
• Design increased by 2.5%
• Hospitality decreased by 2%
In 2020, the total number of students at the college increased by 0.27% to 2 significant figures.
Define, for each course, a variable for the number of students enrolled on that course in 2019.
Using your variables from part (a)(i), write down three equations that model this situation.
By forming and solving a matrix equation, determine how many students were enrolled on each of the three courses in 2019.
A is a 2 by 2 matrix and B is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
AB
A + B
Given that
where a, b and λ are constants,
determine
• the value of λ
• the value of a
• the value of b
Given that
 1 | 1 | 1  |
 0 | sin | cos  |
determine the values of θ for which the matrix M is singular.
A is a 2 by 2 matrix and B is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
AB
A + B
Given that
where a, b and λ are constants,
determine
• the value of λ
• the value of a
• the value of b
Given that
determine the values of θ for which the matrix M is singular.
Given that k ≠4, find, in terms of k, the inverse of the matrix M.
Find, in terms of p, the coordinates of the point where the following planes intersect.
2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2
Find the value of q for which the following planes intersect in a straight line.
4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2
For this value of q, determine a vector equation for the line of intersection.
Given that k ≠4, find, in terms of k, the inverse of the matrix M.
Find, in terms of p, the coordinates of the point where the following planes intersect.
2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2
Find the value of q for which the following planes intersect in a straight line.
4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2
For this value of q, determine a vector equation for the line of intersection.
Also involves Cayley-Hamilton theorem
where k is a constant.
Show that, in terms of k, a characteristic equation for M is given by
Given that det M = 5
find the value of k
use the Cayley‑Hamilton theorem to find the inverse of M.
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