MTH5112 Linear Algebra I 20012–2013
Coursework 11
Please do not submit solutions to this final coursework, as it is not assessed and will not be marked.
Solutions will be available from the course web site.
Exercise 1. Let A∈Rm×n.
(a) Show that N(ATA) = N(A).
(b) Use (a) to show that the columns of Aare linearly independent if and only if ATAis invertible.
Exercise 2. Let Abe a square matrix. Show the following:
(a) 0is an eigenvalue of Aif and only if Ais not invertible;
(b) If Ais invertible, then λis an eigenvalue of Aif and only if λ−1is an eigenvalue of A−1;
(c) If λis an eigenvalue of Aand na positive integer, then λnis an eigenvalue of An.
Exercise 3. Let
A=
4 3 3
−4−3−4
2 2 3
.
(a) Show that (3,−4,2)Tis an eigenvector of Aand find the corresponding eigenvalue.
(b) Determine all eigenvalues of Aand find bases for the corresponding eigenspaces.
(c) Using (b), explain why Ais diagonisable and find a matrix Pthat diagonalises A.
(d) Using (c), find A5.
Exercise 4. For each of the following matrices A, find the eigenvalues and find a basis for the
corresponding eigenspaces. Decide whether the matrix is diagonalisable, and, if it is, find an invertible
matrix Psuch that P−1AP is diagonal:
(a) 2 3
4 1,(b)
6 4 2
−7−6−5
4 4 4
,(c)
2001
0200
0030
0003
,(d)
2000
0200
0031
0003
.
Exercise 5. Let Abe a symmetric matrix.
(a) Show that (Ax)·y=x·(Ay)for any vectors x,y∈Rn.
(b) Use (a) to show that eigenvectors of Acorresponding to distinct eigenvalues are orthogonal.
Exercise 6. Show the following converse of the Spectral Theorem: if a square matrix Ais diago-
nalised by an orthogonal matrix, then Ais symmetric. [Hint: use the fact that a diagonal matrix is
symmetric.]
Exercise 7. For each of the following symmetric matrices Afind an orthogonal matrix that diago-
nalises it, that is, find an orthogonal matrix Qsuch that QTAQ =D, where Dis diagonal:
(a) 1 2
2 1,(b) 2−2
−2 5 ,(c)
022
202
220
.
[Hint for (c): one of the eigenvalues is −2.]
Have a nice break!