By Spiegel M. R.

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For example, if B is nonsingular, then A B = C B implies A = C, 26 MATRIX ALGEBRA since we can multiply on the right by B 1 to obtain ABB-1 = CBB_1, A I = CI, A = C. 5. The inverse of the transpose of a nonsingular matrix is given by the transpose of the inverse: (A')"1 = (A"1)'. 75) If the symmetric nonsingular matrix A is partitioned in the form An »12 a'12 o22 then the inverse is given by A" 1 = - ( b \ bA n + Aji^a'^Aü1 _a i2An -A^au \ ^ 1 ) ' (2 γ6) where b = 0,22 — ai2Aj~11ai2. A nonsingular matrix of the form B + cc', where B is nonsingular, has as its inverse (B + c c ' ) " 1 = B - 1 - ** C C * .

Then some possible products are Ab, c'A, a'b, b'a, and a b ' . For example, let I-It)· H-2 · H ? · - -I OPERATIONS 15 Then Ab c'A c'Ab a'b b'a ab' ac' Note that A b is a column vector, c'A is a row vector, c'Ab is a scalar, and a'b = b'a. The triple product c'Ab was obtained as c'(Ab). The same result would be obtained if we multiplied in the order (c'A)b: (c'A)b = (1 - 19 - 17) I 3 j = -123. This is true in general for a triple product: A B C = A ( B C ) = (AB)C. 28) Thus multiplication of three matrices can be defined in terms of the product of two matrices, since (fortunately) it does not matter which two are multiplied first.

Suppose A i s n x p , a i s p x 1, b is p x 1, and c is n x 1. Then some possible products are Ab, c'A, a'b, b'a, and a b ' . For example, let I-It)· H-2 · H ? · - -I OPERATIONS 15 Then Ab c'A c'Ab a'b b'a ab' ac' Note that A b is a column vector, c'A is a row vector, c'Ab is a scalar, and a'b = b'a. The triple product c'Ab was obtained as c'(Ab). The same result would be obtained if we multiplied in the order (c'A)b: (c'A)b = (1 - 19 - 17) I 3 j = -123. This is true in general for a triple product: A B C = A ( B C ) = (AB)C.