By D. E. Littlewood
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Additional info for A University Algebra: An Introduction to Classic and Modern Algebra
Consider the product o f A with its adjugate matrix. This is, written in full, a ll> a l2> a 13> a i 4 0 21 » a 22> °2 3> ° 2 4 0 3 1 » a 32> ®33> ®34 _ ° 4 1 » tt42> 0 4 3 » 0 4 4 __ -^11» -^21» -^31» ^41 -^12» -^22» -^-32> ^42 ■^13» -^23» ^33» -^43 A i 4, A 24, A 34, A 44_ In the product the first term in the leading diagonal is a l l ^ l l + al 2^12 + a13-^13 + a l4^14 = 1 a3t I = denoting b y A the determinant o f A . The expression is, in fact, the expansion o f the determinant in terms o f the first row, since the alternating signs are absorbed into the minors in forming the signed minors.
1 a:, 0, — x, in 1 2 0_ Oi 0, 3. ~ 3, 1, 0, 2, 1 1. ro, 1, 1, o, 3, 1, _1, 1, - — Z x — Z y, Z y, 2x + y , x 0, + 5y , an 2 ’ — x + y — 2 x — y — x — by 0 The Transform o f a Matrix The set o f linear equations x = axx + bxy + cxz, y = a%c + b $ + c2z, z = azx + 63y + czz, may be regarded as the transformation from one frame o f reference to another concerning the coordinates o f a given point. Alternatively it may be supposed that there is a unique frame o f reference, and that the equations refer to a one-one point correspon dence in which the point P ' == (x\ y', z ) corresponds to the point P = (x , y , z), which is written P —> P '.
From its form, this determinant is both linear and alternating in the rows o f [asJ. Hence it is a multiple o f the deter minant |ast [ by a factor which is independent o f [asi]. Similarly, by consideration o f the columns, it is a multiple o f the determinant |bst | by a factor which is independent o f [bst], Hence it is o f the form k | ||b8t |, where k is independent o f both matrices. Comparing coefficients o f a11a22 a33b11b22b3Z it is clear that k = 1, and 1^116^1 = 1 ^ 6 « ] . The proof is clearly quite general and holds for square matrices o f any order.