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Math 2270 Quiz 3 1. Decide whether the following matrix is invertible and find the inverse if it exists. If you claim the inverse does not exist, tell me why this is the case. 2 3 6 9 Recall that we call a matrix A invertible if we can solve the equation A~x = ~b uniquely for any choice of ~b. Let’s see if that is the case here. .. .. b1 3 1 . 2 3 . b 1 2 2 . ∼ .. .. 0 0 . b − 3b 6 9 . b 2 2 1 From here we can see that the equation A~x = ~b cannot always be solved, in particular when the components of ~b satisfy b2 − 3b1 6= 0. Therefore, the matrix in question is not invertivble. 2. Consider a matrix of the form A= a b b −a , where a2 + b2 = 1 (a reflection about some line L). Find all nonzero vectors ~v such that A~v = ~v in terms of ~u, where ~u is a unit vector parallel to L. The matrix A represents a reflection about a line, and we are looking for a vector that A holds fixed. Obviously, and vector not on the line will be reflected about the line by A and therefore will not be fixed by the transformation. However, if ~v is a vector on L, then its reflection about L is itself, so it is fixed by the transformation. In particular, we have ~v = k~u for some scalar k.