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Homework 2 Due: Wednesday, September 10 1. Write the system of linear equations 2x − 3y + z = 8 x − 2y − 2z = 4 in the following three formats: (a) as an augmented matrix, (b) in matrix-vector format (i.e., Ax = b), and (c) as a linear combination of the three columns of A. 2. Suppose ~v ∈ Rn is a solution to the homogeneous matrix equation Ax = 0. Suppose λ ∈ R. Show that λ~v is also a solution to this equation. 3. Prove part (2) of Theorem MMDAA on page 211 of Beezer: If B, C ∈ Mat m,n (R) and D ∈ Mat n,p (R), then ( B + C ) D = BD + CD. (H INT: Beezer proves part (1), and uses the notation Rm×n for Matm,n (R).) 4. Our three elementary row operations can be captured as multiplication (on the left) by special elementary matrices. For example, for a 3 × 3 matrix, you can swap rows 1 and 3 by multiplying (on the left) by 0 0 1 0 1 0 . 1 0 0 For example: 7 8 9 1 2 3 0 0 1 0 1 0 · 4 5 6 = 4 5 6 1 2 3 7 8 9 1 0 0 (a) What is the elementary matrix corresponding to multiplying row 2 of a 3 × 3 matrix by α? (b) What is the elementary matrix corresponding to adding β times row 3 to row 1 in a 3 × 3 matrix? 5. Consider the matrices A and B: A= Professor Dan Bates Colorado State University 2 4 3 1 B= 1 − 10 3 10 2 5 − 15 M369 Linear Algebra Fall 2008 (a) Demonstrate that A and B are inverses. (b) Solve the following system of linear equations by writing it in matrix-vector form and using B appropriately: 2x + 4y = 3 3x + y = 1 (Notice that you can check your answer using A.) 6. For each of the following matrices, does the matrix have an inverse? If so, find it. If not, why not? 1 0 3 (a) 2 1 8 1 2 7 2 0 4 (b) 0 6 0 6 0 8 Professor Dan Bates Colorado State University M369 Linear Algebra Fall 2008