assignment1 - School of Engineering Science ENSC 483 Modern...

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School of Engineering Science ENSC 483 - Modern Control Systems Spring 2007–Assignment I (Warm-up Homework!) 1. Use Laplace’s Expansion to evaluate the determinants of the following matrices: A = 3 0 1 2 4 3 1 1 2 ; B = 1 2 3 4 2 3 4 5 3 4 5 6 4 5 6 7 2. Find the inverse of matrices in problem 1. 3. (a) You are told that matrices A , and B are symmetric and that their product is symmetric. What can you conclude from this? (b) A matrix is called idempotent if A 2 = A . Consider the case of 2 × 2 idempotent matrices and say as much as you can about them, and based on your arguments give an example of a 2 × 2 idempotent matrix. 4. Show that the set of all real n × n matrices with usual operation of matrix addition and the usual operation of multiplication of matrices by scalars constitutes a vector space over the reals (i.e., < n × n , < ). Determine dimension and a basis for this space. Is the above statement still true if ( < n × n , < ) is replaced by ( < m × n , <
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