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PHYS2370 Maths 3 Assignment 3
1. Matrices A, B, C and D are defined by
Some of the following expressions define valid matrices, but others do not. In each case, either work out the specified matrix, or say why the expression is invalid:
A + B, AB, BA, C + D, CD, DC, DA, AD
2. (a) Find the determinants of the two matrices
Compare your two answers, and explain what you find.
(b) Use the results of part (a) to find the determinants of the following matrices, explaining your reasoning in each case:
3. Solve the simultaneous equations
x − y + 2z = 5
−x + 2y + z = 8
2x − y + z = −1
by finding the inverse of a suitable matrix.
4. Relative to a Cartesian coordinate system that she finds convenient, an astronomer has determined the equation of an asteroid’s elliptical orbit as
0.084x2 − 0.079xy + 0.107y2 = 1.
[Distances are measured in astronomical units (1AU is approximately the mean distance of the earth from the sun), and the units of the numerical coefficients are AU−2 .]
Find the semi-major and semi-minor axes of this ellipse, and a unit vector in the direction of each axis.