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N! det (A) = sgn (r1 , · · ·, rn ) sgn (k1 , · · ·, kn ) ar1 k1 · · · arn kn . (r1 ,···,rn ) (k1 ,···,kn ) This proves the corollary since the formula gives the same number for A as it does for AT . 7 If two rows or two columns in an n × n matrix, A, are switched, the determinant of the resulting matrix equals (−1) times the determinant of the original matrix. If A is an n×n matrix in which two rows are equal or two columns are equal then det (A) = 0. Suppose the ith row of A equals (xa1 + yb1 , · · ·, xan + ybn ).

Also suppose A ∈ L (V, V ) . 1 The characteristic polynomial of A is defined as q (λ) ≡ det (λ id −A) where id is the identity map which takes every vector in V to itself. The zeros of q (λ) in C are called the eigenvalues of A. 2 When λ is an eigenvalue of A which is also in F, the field of scalars, then there exists v = 0 such that Av = λv. 26. Since λ ∈ F, λ id −A ∈ L (V, V ) and since it has zero determinant, it is not one to one so there exists v = 0 such that (λ id −A) v = 0. The following lemma gives the existence of something called the minimal polynomial.

8. Let L ∈ L (V, W ) and let M ∈ L (W, Y ) . Show rank (M L) ≤ min (rank (L) , rank (M )) . 9. Let M (t) = (b1 (t) , · · ·, bn (t)) where each bk (t) is a column vector whose component functions are differentiable functions. For such a column vector, T b (t) = (b1 (t) , · · ·, bn (t)) , define T b (t) ≡ (b1 (t) , · · ·, bn (t)) Show n det (M (t)) = det Mi (t) i=1 where Mi (t) has all the same columns as M (t) except the ith column is replaced with bi (t). 10. Let A = (aij ) be an n × n matrix. Consider this as a linear transformation using ordinary matrix multiplication.