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.

Answered by Cameron B. Maths tutor

2779 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Integrate the function f(x)=3^x+2 with respect to x


Differentiate y=(4x - 5)^5 by using the chain rule.


1. The curve C has equation y = 3x^4 – 8x^3 – 3 (a) Find (i) d d y x (ii) d d 2 y x 2 (3) (b) Verify that C has a stationary point when x = 2 (2) (c) Determine the nature of this stationary point, giving a reason for your answer.


Calculate the volume obtained when rotating the curve y=x^2 360 degrees around the x axis for 0<x<2


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences