Vectors and matrices: Problem sheet 4

Please answer all questions, and check your answers whenever possible Solutions should be submitted before or on Thursday 2 December 2004

1. Give examples of matrices A and B where

(a) AB exists but BA does not.

(b) AB and BA both exist but have different sizes.

2. Find, whenever possible, the inverses of the following matrices:

( \begin{matrix}3 & −1 \\ 2 & 1 \end{matrix} ) , ( \begin{matrix}1 & i \\ i & 1 \end{matrix} ) , ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) , ( \begin{matrix}1 & 1 & 1 \\ 4 & 3 & 5 \\ 2 & 1 & 3 \end{matrix} ) ,

( \begin{matrix}3 & 1 & 1 \\ 3 & 1 & 2 \\ 1 & 0 & −3 \end{matrix} ) , ( \begin{matrix}1 & 1 & 2 \\ 3 & −5 & −1 \\ −1 & 2 & 1 \end{matrix} ) , ( \begin{matrix}1 & −2 & −1 & 0 \\ 1 & 2 & 1 & 1 \\ 3 & 2 & 2 & 1 \\ 1 & 5 & 2 & 2 \end{matrix} ) .

(In the second example, i^{2} = −1.)

3. Let

A = ( \begin{matrix}3 & 2 \\ −1 & 1 \end{matrix} ) .

Find all matrices

X = ( \begin{matrix}x & y \\ z & t \end{matrix} )

which satisfy AX = XA.

4. For each of the following matrices A, calculate A^{2}, A^{3} and A^{4}. Guess (and if possible prove) a general formula for A^{n}:

( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) , ( \begin{matrix}x & 1 \\ 0 & x \end{matrix} ) , ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{matrix} ) .

5. Let L be the line y = x tan θ in the plane (so that L makes an angle of θ with the x-axis). Find the matrix which represents reflection through the line L. (Hint: where do the points (1, 0) and (0, 1) get reflected to?)

6. Find the equation of the image of the unit circle x^{2} + y^{2} = 1 under the transformation with matrix

( \begin{matrix}2 & 3 \\ 1 & 2 \end{matrix} ) .

7. Which of the following matrices are elementary? For each elementary matrix among them, write down its inverse.

( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ,

( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 & 0 \\ 1 & 1 & 2 \\ 0 & 0 & 1 \end{matrix} ) .

8. True or false? Justify your answers.

(a) If A and B are invertible matrices of the same size then AB has inverse A^{−1}B^{−1}.

(b) If A is a square matrix then (I − A)^{2} = I^{2} − 2A + A^{2}.

(c) If A and B are square matrices of the same size then A^{2} − B^{2} = (A + B)(A − B).

(d) If A and B are square matrices of the same size then (AB)^{2} = A^{2}B^{2}.

(e) If A and B are symmetric matrices of the same size then A + B is symmetric.

(f) If A and B are symmetric matrices of the same size then AB is symmetric.

9. An orthogonal matrix A is a square matrix which satisfies AA^{t} = I (where A^{t} denotes the transpose of A). Prove that if A and B are orthogonal matrices of the same size then AB is also orthogonal.

Verify that the 2-by-2 rotation matrices and reflection matrices (as in question 5) are orthogonal. Can you find any other orthogonal 2-by-2 matrices?

RJC 18/11/2004