Mathematical Methods and Mechanics

Question I

Consider the following simultaneous linear algebraic equations,

g*4x2*3xt=3 ' .rt-9xz-24=-7

3xt-xz*44=k

where rLis au unspecified cofftarit

(a) By usrng the rnethod of Gauss elimination with essential row interehoryes anly, reduce the augmented mafirix to upper tiangulu form-

(9ma*s)

(b) Hencg fmd &e valu{s) of & for which the equations re consistmt, explaining the reasons foryour choice.

(3 ma*s)

@cstion2

(a) Find the eige,nvalus and its comesponding eigenvectors of alx?. marix given by

(9ma*s)

O) Hence, findthe general soltrtion ofthe systemof diffenential equations

*r=4x1 + 6xz *2 = 4x1 +llxz.

(3 ma*s)

l-+ 6l L-, r rl'