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Midterm_Review_Problems

# Midterm_Review_Problems - Math 115 Midterm Review Problems...

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Math 115 Midterm Review Problems 1. (a) Solve the following linear system: x + y + 2 z = 2 2 x + y - z = 3 x + 2 y + 7 z = 3 (b) Find the solution to the associated homogeneous system: x + y + 2 z = 0 2 x + y - z = 0 x + 2 y + 7 z = 0 2. Suppose A , B and C are invertible n × n matrices. Prove the following, using the definition of matrix inverse: ( A - 1 BC ) - 1 = C - 1 B - 1 A 3. Find the values of the number c such that 1 c 0 2 0 c c - 1 1 has an inverse. 4. Find the inverse of 1 - 1 - 2 - 1 0 1 2 1 0 , and use it to solve the system x - y - 2 z = 3 - x + z = 0 2 x + y = 1 5. Consider the transformation T : R 2 R 2 defined as follows: Counterclockwise rotation about the origin through an angle of π/ 2 followed by reflection in the line y = x. Determine the standard matrix for T and find T 1 2 . 6. If T A : R 2 R 2 is a linear transformation defined by T A x y = x + y x - y and T B : R 2 R 2 is a linear transformation defined by T B x y = 3 x 2 x + 4 y , find (a) the standard matrix for T A T B .
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