[SOLVED] SIT292 Assignment 2

1. Find all the cofactors C_ij of

hence find adjA.

(ii) Verify that the adj A you obtained is correct by multiplying it with A. 20 marks

2. Find all numbers α such that the vectors

 5               α 

 α              −α 

 3α  and  3α _ are orthogonal. 

              −1 

 1            

• 1
• marks

3. Use Gaussian elimination to reduce the following system of equations torow-echelon form, hence solve for x₁,…,x₄. Justify the correctness of your solution using matrix ranks

2x₁ − 2x₂ + 2x₃ − 4x₄ = 2 x₁ + 4x₂ + 8x₃ + 2x₄ = 5

−x₁ + 9x₂ + 3x₃ − 4x₄ = 5

4. Use Gaussian elimination to reduce the following system of equations torow-echelon form, hence solve for x₁,…,x₃. Justify the correctness of your solution using matrix ranks

3x₂ + 11x₃ = 6 x₁ + x₂ + 3x₃ = 2

3x₁ − 3x₂ − 13x₃ = −6

−x₁ + 2x₂ + 8x₃ = 4

5. Use Gaussian-Jordan elimination to find the inverse of the matrix

6. Find eigenvalues and eigenvectors of the matrices A and B, where

and .

Hint: You can guess one eigenvalue for A,B and use long division of polynomials to find the others. 20 marks

7. Diagonalise the matrix

 .

8. For the matrices
   • find the eigenvalues and eigenvectors
   • determine, if possible, a matrix P so that P⁻¹AP = B. If impossible, provide an argument for that. (Hint: Use the method in the Study Guide p.113).

9. For the following matrix
   • find the eigenvalues
   • for each eigenvalue determine the eigenvector(s)
   • determine a matrix P so that B = P⁻¹AP is in triangular form, and verify that the determinant of B agrees with what you used in (a)

( 5 + 5 + 10 = 20 marks)