# Practice Problems 5

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Published on March 6, 2014

Author: anhtuantran509

Source: slideshare.net

Mathematical Methods for Quantitative Finance 1. Consider the matrix Practice Problems 5   1 1 0 A =  1 5 2 −1 1 1 (a) Use elimination to turn A into an upper triangular matrix. How many pivots does A have? (b) Let b = (2, 8, 3). Does Ax = b have a solution? (c) Let b = (2, 8, 1). Does Ax = b have a solution? (d) Can you ﬁnd multiple solutions in either part (b) or part (c)? If so, ﬁnd 2. (e) Does A have an inverse? Justify your answer using results from this exercise. 2. Let AB = I and CA = I where I is the n × n identity matrix. (a) What are the dimensions of the matrices A, B and C? (b) Show that B = C. (c) Is A invertible? [Hint: you can write B = IB] 3. Let A be a square matrix with the property that A2 = A. Find expressions for (I − A)2 and (I − A)5 . 4. (a) Can the vector (9, 2, −5) be written as a linear combination of the vectors (1, 2, 3) and (6, 4, 2)? (b) Does a system of equations with coeﬃcient matrix   1 6 9 2 A = 2 4 3 2 −5 have a full set of pivots? http://computational-finance.uw.edu

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