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1. MATRICES 45 Proof: (AIn )ij = Aik δ kj k = Aij and so AIn = A. The other case is left as an exercise for you. 21 An n × n matrix, A has an inverse, A−1 if and only if AA−1 = A−1 A = I where I = (δ ij ) for 1 if i = j δ ij ≡ 0 if i = j Such a matrix is called invertible. 4 Finding The Inverse Of A Matrix A little later a formula is given for the inverse of a matrix. However, it is not a good way to find the inverse for a matrix. There is a much easier way and it is this which is presented here.

Because Ω (t) × u (t) ≡ i∗ w1 u1 j∗ w2 u2 k∗ w3 u3 ≡ i∗ w2 u3 − w3 u2 + j∗ w3 u1 − w13 + k∗ w1 u2 − w2 u1 . This proves the lemma and yields the existence part of the following theorem. 4 Let i (t) , j (t) , k (t) be as described. Then there exists a unique vector Ω (t) such that if u (t) is a vector whose components are constant with respect to i (t) , j (t) , k (t) , then u (t) = Ω (t) × u (t) . Proof: It only remains to prove uniqueness. Suppose Ω1 also works. Then u (t) = Q (t) u and so u (t) = Q (t) u and Q (t) u = Ω×Q (t) u = Ω1 ×Q (t) u for all u.

You may need to study these later. 1. 2 Matrices which are n × 1 or 1 × n are especially called vectors and are often denoted by a bold letter. Thus   x1   x =  ...  xn is a n × 1 matrix also called a column vector while a 1 × n matrix of the form (x1 · · · xn ) is referred to as a row vector. All the above is fine, but the real reason for considering matrices is that they can be multiplied. This is where things quit being banal. First consider the problem of multiplying an m × n matrix by an n × 1 column vector.

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