LINEAR ALGEBRA
1. 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
2. Find all numbers α such that the vectors
5 α
α −α
3α and 3α are orthogonal.
−1
1
5 1
6 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
10 marks
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
10 marks
5. Use Gaussian-Jordan elimination to find the inverse of the matrix
10 marks
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
.
10 marks
8. For the matrices
(a) find the eigenvalues and eigenvectors
(b) 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).
( 10 + 4 = 14 marks)
9. For the following matrix
(a) find the eigenvalues
(b) for each eigenvalue determine the eigenvector(s)
(c) 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)
