Exercise 13 Determinant via ERO and Elementary Matrices Use the method of ERO

# Exercise 13 determinant via ero and elementary

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Exercise 13.[Determinant via ERO and Elementary Matrices]Use the method of ERO and Elementary Matrices, find the determinant of theBmatrixin Exercise12[ det(B) =-168 ] KVL/Aug16
EE2007 Tutorial 4 Linear Algebra Linear Combination, Independence, Span, Basis Exercise 14.[Concepts and Examples discussed in lectures]The following were discuss in lectures.Review them and ask questions in the tutorialclass, if necessary. Generate your own example to test your understanding.Givenv,v1,v2andv3inR3. What do you have to do to(a) Determine whethervis a linear combination ofv1,v2andv3(b) Findspan(v1,v2) andspan(v1,v2,v3)(c) Determine whethervis inspan(v1,v2,v3)(d) Determine whetherv1,v2,v3are linearly independent or not.Exercise 15.[Linear Combinations](a) LetA=12-1234-321andx=-1-13.Write the productAxas a linearcombination of the column vectors ofA.(b) LetA=-1-234andB=3225.Write each column vector ofABas alinear combination of the column vectors ofA.(c) Describe all vectors inR3that can be written as a linear combination of the vectors12-1,37-2,130[ [a, b, c]0such that 3a-b+c= 0 ]Exercise 16.[Row and column spaces of a matrix]Consider the matrixA=1-1013-3.KVL/Aug16
1 2 3 in the column space of A ? (c) Describerow(A) andcol(A). Exercise 17.[Rank and nullity]IfAis a 3×5 matrix, explain why the columns ofAmust be linearly dependent. Whatare the possible values ofnullity(A)? KVL/Aug16
EE2007 Tutorial 5 Linear Algebra Diagonalisation, Eigenvalues and Eigenvectors Exercise 18.[Diagonalisation of matrix]Explain how eigenvalues and eigenvectors are used to diagonalise an×nmatrix. Whatcondition(s) is(are) needed so that a matrix A can be diagonalised? The next two exercisesreinforce the concept.[ see lecture notes ]Exercise 19.[Diagonalisation of matrix]LetA=00202000-1(a) Find the eigenvalues ofA.(b) From your result in part (a) can you conclude whetherAis diagonalizable? Explain.(c) Find the eigenvectors corresponding to each eigenvalue.(d) Are the eigenvectors found in part (c) linearly independent? Explain.(e) From your result in part (d) can you conclude whetherAis diagonalizable? Explain.(f) If your answer to part (e) is yes, find a matrixPthat diagonalizesA. Specify thediagonal matrixDsuch thatD=P-1AP.