By Bernard Kolman
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The vec tor (b) B has a row consisting entirely of zeros? 43. jf A = [aii ] is an II x II . matrix, then the Irace of A. Tr(A). is defined as the sum of all elements on the main diagolwl of A. Tr(A) = LUil Show each of the follow;=1 mg: (a) Tr(cA) = c Tr(A). where c is a real number (b) T r(A + B) = Tr(A) + Tr ( B ) (e) Tr(AB) = Tr(BA) (d ) Tr(AT) = Tr(A) (e) Tr(A T A) 2: 0 44. Compute the trace (see Exercise 43) of each of the following matrices: (a) Ie) gives the price (in dollars) of each receiver, CD player, speaker.
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9 EXAMPLE 3 EXAMPLE 4 • A matri x A with real entries is called symmetric if A T = A. A matrix A with real e ntries is called s kew symmetric if A T = - A. A ~ B ~ Un 2 4 S [-~ -n, , 2 0 - 3 • is a symmetric matri x. • ' kew 'ymmell'e mo,,'x. 4 We can make a few observations about symmetric and skew symmetric matrices; the proofs of most of these statements will be left as exercises. It follows from thc precedi ng definitions that if A is symmetric or skew ~y m metric, then A is a square matrix.