Description
- Find all the cofactors Cij of
hence find adjA.
(ii) Verify that the adj A you obtained is correct by multiplying it with A. 20 marks
- Find all numbers α such that the vectors
 5              α 
 α             −α 
 3α  and  3α  are orthogonal. 
             −1 
 1           
- 1
- marks
- 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
- 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
- Use Gaussian-Jordan elimination to find the inverse of the matrix
- 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
- Diagonalise the matrix
 .
- 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).
- 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)