FURTHER MATHEMATICAL TECHNIQUES (MA10193) EXAMPLES SHEET 2

I, or somebody, will look at work given to me at the end of the examples class at 12:15 on Wednesday, 9th 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. Solve the following set of simultaneous equations using elementary row operations:

\begin{aligned}x + 2y − 8z & = 0 \\ 2x − 3y + 5z & = 0 \\ 3x + 2y − 12z & = 0 \end{aligned}

2. Use the method of Gauss-Jordan elimination to solve the following set of simultaneous equations:

\begin{aligned}9x + 4y + 3z & = −1 \\ 5x + y + 2z & = 1 \\ 7x + 3y + 4z & = 1 \end{aligned}

3. Use elementary row transformations to find the solution to the following simultaneous equations in the case where a_{11}a_{22} − a_{12}a_{21} ≠ 0.

\begin{aligned}a_{11}x_{1} + a_{12}x_{2} & = b_{1} \\ a_{21}x_{1} + a_{22}x_{2} & = b_{2} \end{aligned}

4. Find the rank of A where

A = ( \begin{matrix}0 & 2 & −1 \\ 1 & 0 & 3 \\ 2 & −4 & 2 \end{matrix} ) .

Hence (without solving the system) state how many solutions, if any, the following linear system has:

\begin{aligned}2y − z & = 1 \\ x + 3z & = 3 \\ 2x − 4y + 2z & = −7 \end{aligned}

5. Solve the equation (remember that det A or |A| means the determinant of A)

| \begin{matrix}x + 1 & x & x − 4 \\ 2 & 1 & −4 \\ 3 & 5 & 1 \end{matrix} | = 0

6. Consider the matrix

A = ( \begin{matrix}−1 & 3 & x \\ −2 & 2 & 1 \\ x & 4 & 0 \end{matrix} ) .

Calculate det A in terms of x and hence determine those values of x (to 2 d.p.) for which the matrix has rank less than 3. If x = 2 write down the rank of A.

GKS, 2/03/05