MATHEMATICS 2 (MA10193) EXAMPLES SHEET 1

I, or somebody, will look at work given to me at the end of the examples class at 12:15 on Wednesday, 2nd March, or left before then in the folder on my office door (1W3.35). If you do not have a copy of this sheet, you can find one at

http://www.bath.ac.uk/~masgks/MA10193/sheet1.ps (or .dvi or .pdf).

1. Suppose that the matrix

A = ( \begin{matrix}cos w & p \\ sin w & q \end{matrix} )

satisfies the equation AA^{T} = I. Find the values of p and q.

2. Check, by calculation, that (AB)^{T} = B^{T} A^{T}, where

A = ( \begin{matrix}1 & 1 & 2 & −1 \\ 3 & 2 & 1 & −2 \\ 0 & 2 & 1 & 1 \end{matrix} ) and B = ( \begin{matrix}2 & 3 \\ −1 & 2 \\ 3 & 0 \\ 1 & 1 \end{matrix} ) .

3. Expand (A + B)^{2}. Check that your expansion is correct in the case where

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

4. Suppose that

A = ( \begin{matrix}1 & x & 1 \\ x & 2 & y \\ 1 & y & 3 \end{matrix} ) and B = ( \begin{matrix}3 & −3 & z \\ −3 & 2 & −3 \\ z & −3 & 1 \end{matrix} ) ,

and AB is symmetric. Is it true that AB = BA in this case? Find x, y and z (this is a bit harder).

5. Let

B = ( \begin{matrix}4 & 1 & −2 \\ 2 & −1 & 3 \\ 3 & 2 & 1 \end{matrix} ) .

Represent B as the sum of a symmetric matrix and a skew-symmetric matrix.

6. Let

A = ( \begin{matrix}2 & 5 & 1 \\ 1 & −1 & 2 \\ 0 & 2 & 3 \end{matrix} ) , B = ( \begin{matrix}2 & 1 \\ −3 & 0 \\ 1 & 4 \\ 3 & 2 \end{matrix} ) and C = ( \begin{matrix}2 \\ 1 \\ 3 \end{matrix} ) .

Say which of the following expressions are defined, and calculate those that are:

a) A^{2}, B^{2}, C^{2}.

b) AA^{T}, B^{T} B, CC^{T}, C^{T} C.

c) C^{T} AB, (AB)^{T} C.

7. Suppose that

A = ( \begin{matrix}1 & 0 & 2 \\ 2 & −1 & 3 \\ 4 & 1 & 8 \end{matrix} ) and B = ( \begin{matrix}−11 & 2 & 2 \\ −4 & 0 & 1 \\ 6 & −1 & −1 \end{matrix} ) .

a) Verify that AB = BA = I.

b) Use part (a) and the rules about multiplication and transposes of matrices to write down the value of A^{T} B^{T}.

c) Use (a) and (b) to write each of the following systems of equations in matrix form and thereby solve them:

\begin{aligned}−11x + 2y + 2z & = 1 \\ −4x + z & = 2 \\ 6x − y − z & = 3 \end{aligned}

and

\begin{aligned}−11x − 4y + 6z & = 1 \\ 2x − z & = 2 \\ 2x + y − z & = 3. \end{aligned}

GKS, 22/02/05