A matrix M has eigenvectors (3,1,0) (2,8,2) (1,1,6) with corresponding eigenvalues 1, 6, 2 respectively. Write an invertible matrix P and diagonal matrix D such that M=PD(P^-1), hence calculate M^5.

Without even knowing M, the candidate can calculate M^5. This will follow from the fact that P is the matrix consisting of the eigenvectors of M as columns, and D will have the eigenvalues (in matching columns to their corresponding eigenvectors) down the lead diagonal. The candidate will have to do some computation to determine P^-1, but this is standard in A-level and will serve as good practice.Then we see that M^5 = (PD(P^-1))^5 = P(D^5)(P^-1), the essence behind this being that D^5 is very simple to calculate since D is diagonal.Again this final stage requires some computation, but getting comfortable with this serves as a great means to reduce the pressure of time in the exam.

CB
Answered by Cameron B. Maths tutor

3096 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

The curve C has equation: 2x^2y + 2x + 4y – cos (piy) = 17. Use implicit differentiation to find dy/dx in terms of x and y.


Integrate tan (x) with respect to x.


Differentiate f(x) = (3x + 5)(4x - 7)


Differentiate e^(xsinx)


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning