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**Unformatted text preview: **MATH 152 L1 & L2 Applied Linear Algebra and Diﬀerential Equations Week 09 Worksheet (April 4, 2005): Systems of Linear Equations Q1. (Method of reduction) Solve the system of linear equations x1 3x1 2x1 −2x2 −6x2 −4x2 Spring 2004-05 +2x3 +7x3 +3x3 = = = 0, −1, 1. Q2. (Method of reduction) Solve the system −2x1 +4x2 x1 −2x2 of linear equations −x3 −5x3 +3x3 −3x4 +7x4 −2x4 +2x5 +x5 +x5 = = = 2, 1, 0. Q3. (Existence and uniqueness) Given 3 −6 2 −1 −2 4 1 3 , A= 0 0 1 1 1 −2 1 0 b1 b 2 , b= b3 b4 and x y . x= z w Under what conditions on b1 , b2 , b3 , b4 so that the linear system Ax = b has solutions. Q4. (Existence and uniqueness) Given a −8 a A = 3 2 2 −1 1 , 1 and x x = y . z Under what condition on a (here a is a real number) so that the homogeneous system Ax = 0 has solutions other than the trivial solution x = 0? Q5. (Pivots and rank) Find an example of 3 × 3 invertible matrices A and B, such that rank (A + B) = 1 (so that A + B is not invertible). Find an example of 3 × 3 non-invertible matrices A and B, such that A + B is invertible. 1 Q6. (Pivots and rank) Given the matrix A = 2 1 of A is 2. λ −1 10 −1 λ −6 2 5. Find value(s) of λ such that the rank 1 Q7. (Pivots and rank) Suppose Ax = b is a consistent system of 4 linear equations in 8 variables. If there are 6 free variables in the general solution, what is the rank of A? 1 Q8. (Find inverse of matrix) Consider the matrix A = 1 1 (a) Find the inverse of A by row operations. 0 1 1 0 0 . 1 (b) Find the inverse of At , i.e., the inverse of the transpose of A. (c) Solve the matrix equation AXAt = At − 2A + I, where X is an unknown 3 × 3 matrix and I is the 3 × 3 identity matrix. ...

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