Practice Midterm 1 (Vojta) - Solutions - Math 54 Solutions to Sample First Midterm 1 1(10 points Find the inverse of the matrix A = 2 1 algorithm

Practice Midterm 1 (Vojta) - Solutions - Math 54 Solutions...

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Math 54. Solutions to Sample First Midterm 1. (10 points) Find the inverse of the matrix A = 1 2 3 2 6 7 1 1 2 , if it exists. Use the algorithm introduced in Chapter 2. The algorithm uses row reduction of the matrix [ A I ] : 1 2 3 1 0 0 2 6 7 0 1 0 1 1 2 0 0 1 1 2 3 1 0 0 0 2 1 - 2 1 0 0 - 1 - 1 - 1 0 1 1 2 3 1 0 0 0 - 1 - 1 - 1 0 1 0 2 1 - 2 1 0 1 2 3 1 0 0 0 - 1 - 1 - 1 0 1 0 0 - 1 - 4 1 2 1 2 0 - 11 3 6 0 - 1 0 3 - 1 - 1 0 0 - 1 - 4 1 2 1 0 0 - 5 1 4 0 - 1 0 3 - 1 - 1 0 0 - 1 - 4 1 2 1 0 0 - 5 1 4 0 1 0 - 3 1 1 0 0 1 4 - 1 - 2 Therefore the inverse is - 5 1 4 - 3 1 1 4 - 1 - 2 . 2. (10 points) A matrix A and an echelon form of A are given here: A = 1 2 - 1 1 - 1 - 2 - 4 3 - 3 0 1 2 - 3 3 3 1 2 - 2 2 1 1 2 - 1 1 - 1 0 0 1 - 1 - 2 0 0 0 0 0 0 0 0 0 0 . (a). Write the solution set of the homogeneous system A~x = ~ 0 in parametric vector form (i.e., as a linear combination of fixed vectors, in which the weights are allowed to take on arbitrary values).
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