SIT292 LINEAR ALGEBRA ASSIGNMENT

Added on - 25 Nov 2019

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SIT292 LINEAR ALGEBRA2017ASSIGNMENT 2STUDENT ID[Pick the date]
Question 1(i)Given matrixA=[321112120]CofactorsCijof the matrixCij=¿C11=¿C12=¿C13=¿C21=¿C22=¿C23=¿C31=¿C32=¿C33=¿Hence, cofactors of the matrix A would be given below:1
CofactorsA=[421214351]AdjAof the matrix A would be given below:AdjA=[423215141](ii)Verification that computedadjAis correct.IfA(adjA)detA=INow,LHSA(adjA)=[321112120].[423215141]¿[3.4+2.(2)+1.(1)3.(2)+2.1+1.43.3+2.(5)+1.11.4+1.(2)+2.(1)1.(2)+1.1+2.41.3+1.(5)+2.11.4+(2).(2)+0.(1)(1).(2)+(2).1+0.4(1).3+(2).(5)+0.1]¿[700070007]AndA=[321112120]2
detA=|321112120|¿3(0+4)2(0+2)+1(2+1)¿1241¿7A(adjA)detA=[700070007]7=[100010001]RHSI =[100010001]LHS = RHSIt is apparent that both the sides are equal and therefore, the computedadjAis correct.3
Question 2Two vectors would be orthogonal when there dot product becomes zero.¿7+.+3.(3)+1.1+(4).4¿7+29+116¿2215Hence,2215=0(5)(+3)=0=5,3Therefore, the for=5,3thevectorswouldbeorthogonal.4
Question 3Given equationsGaussian elimination method to reduce the following system of equations into row echelon formis applied below:5
The row echelon form of the given system is given below:This system does not have any solution because05.It is apparent from the above that the given system has three linear equations and 4 variables.Hence, the system is said to be inconsistent and does not have any solution as evident from theabove.6
Question 4Eigen values and eigenvectors of the given matrixes are as highlighted below:For matrix AEigen valuesA=[101010101]Let λ is the eigenvalues of the given matrix in such a way thatdet(A¿λI)=0¿Now,AλI=[101010101]λ[100010001]AλI=[101010101][|λ000λ000λ|]AλI=[1λ0101λ0101λ]Determinant ofAλI7