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SIT292 LINEAR ALGEBRA2017ASSIGNMENT 2STUDENT ID[Pick the date]
Question 1(i)Given matrixA=[321112−1−20]CofactorsCijof the matrixCij=¿C11=¿C12=¿C13=¿C21=¿C22=¿C23=¿C31=¿C32=¿C33=¿Hence, cofactors of the matrix A would be given below:1
CofactorsA=[4−2−1−2143−51]AdjAof the matrix A would be given below:AdjA=[4−23−21−5−141](ii)Verification that computedadjAis correct.IfA∗(adjA)detA=INow,LHSA∗(adjA)=[321112−1−20].[4−23−21−5−141]¿[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)+2.1−1.4+(−2).(−2)+0.(−1)(−1).(−2)+(−2).1+0.4(−1).3+(−2).(−5)+0.1]¿[700070007]AndA=[321112−1−20]2
detA=|321112−1−20|¿3(0+4)−2(0+2)+1(−2+1)¿12−4−1¿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∝+∝2−9∝+1−16¿∝2−2∝−15Hence,∝2−2∝−15=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 because0≠−5.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,A−λI=[101010101]−λ[100010001]A−λI=[101010101]−[|λ000λ000λ|]A−λI=[1−λ0101−λ0101−λ]Determinant ofA−λI7
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