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mat22a-practiceexam2-sol

mat22a-practiceexam2-sol - Math 22A UC Davis Winter 2011...

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Math 22A UC Davis, Winter 2011 Prof. Dan Romik Solutions to practice question set 2 1. (a) Define A = 1 3 - 1 1 4 - 1 - 1 - 3 2 . Compute the inverse matrix of A . Solution. We follow the standard method of computing the inverse matrix by performing a sequence of elementary row operations trans- forming the matrix into the identity matrix, and at the same time per- forming the same sequence of operations on the identity matrix: 1 3 - 1 1 0 0 1 4 - 1 0 1 0 - 1 - 3 2 0 0 1 R 2 - R 1 ,R 3 + R 1 1 3 - 1 1 0 0 0 1 0 - 1 1 0 0 0 1 1 0 1 R 1 - 3 R 2 1 0 - 1 4 - 3 0 0 1 0 - 1 1 0 0 0 1 1 0 1 R 1 + R 3 1 0 0 5 - 3 1 0 1 0 - 1 1 0 0 0 1 1 0 1 Thus we get the answer A - 1 = 5 - 3 1 - 1 1 0 1 0 1 . Remarks 1. An alternative method for computing A - 1 is by using the formula A - 1 = 1 det( A ) adj( A ). 2. In questions like this it is highly advisable to verify your answer by checking that AA - 1 = I . (b) Find the solution v = x y z to the equation Av = 2 - 1 0 .
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