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Further Mathematics

A Level

Let A, B and C be nxn matrices such that A=BC-CB. Show that the trace of A (denoted Tr(A)) is 0, where the trace of an nxn matrix is defined as the sum of the entries along the leading diagonal.

This is the type of question which requires some out of the box thinking. A sketch is presented below: We start by defining A = (a_ij), B = (b_ij) and C = (c_i...

Answered by Daniel G. • Further Mathematics tutor

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Express sin(5theta) in terms of sin(theta) and powers of sin(theta) only.

Consider the expression (cos(theta) + i*sin(theta))⁵. (Where theta a real parameter).By De Moirve's theorem, we know this expression is equivalent to cos(5theta) + i sin(5theta).We can also ap...

Answered by Peter F. • Further Mathematics tutor

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A mass m=1kg, initially at rest and with x=10mm, is connected to a damper with stiffness k=24N/mm and damping constant c=0.2Ns/mm. Given that the differential equation of the system is given by d^2x/dt^2+(dx/dt *c/m)+kx/m=0, find the particular solution.

The system is described by a homogeneous, second order differential equation d²x/dt²+(dx/dt * c/m) + kx/m =0. First, substitute the known constants (m,k,c) to get d²x/dt<...

Answered by Christodoulos K. • Further Mathematics tutor

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Find the eigenvalues and eigenvectors of the following 3x3 matrix (reading left to right, top to bottom): (1 0 2 3 1 1 2 0 1)

The eigenvalues are given by the characteristic equation (1-x)((1-x)^2-4)=0, which gives the values x=1, x=-1 and x=3 . These eigenvalues correspond to the eigenvectors (0, 1, 0), (1, -1, -1) and (1, -...

Answered by Joshua P. • Further Mathematics tutor

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