By Smith S.B.

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Thus the ij-entry of BT AT is b1j ; b2j ; . . ; bmj ai1 ; ai2 ; . . ; aim T b1j ai1 b2j ai2 . . bmj aim Thus ABT BT AT , since the corresponding entries are equal. 14. Find the diagonal and trace 2 3 of each matrix: 2 1 3 6 2 (a) A 4 2 À5 8 5, (b) B 4 3 4 À2 9 À5 (a) 4 À7 0 3 8 9 5, 2 C 1 2 4 À5 À3 . 6 The diagonal of A consists of the elements from the upper left corner of A to the lower right corner of A or, in other words, the elements a11 , a22 , a33 . Thus the diagonal of A consists of the numbers 1; À5, and 9.

Namely, tr A a11 a22 a33 . . ann The following theorem applies. 7. Suppose A aij and B bij are n-square matrices and k is a scalar. Then: tr AT tr A, (i) tr A B tr A tr B, (iii) (ii) (iv) tr AB tr BA. 6. 4, tr AT tr A; tr A B tr A tr B; tr 2A 2 tr A Furthermore, although AB T BA, the traces are equal. Identity Matrix, Scalar Matrices The n-square identity or unit matrix, denoted by In, or simply I, is the n-square matrix with 1's on the diagonal and 0's elsewhere.

Then A2 , which is diagonal. 24. Find an upper triangular matrix A such that A 0 3 À57 . 27 x y Set A . Then x3 8, so x 2; and z3 27, so z 3. Next calculate A3 using x 2 and 0 z y 3: 2 y 2 y 2 y 4 5y 4 5y 8 19y A2 and A3 0 3 0 3 0 3 0 9 0 9 0 27 2 À3 Thus 19y À57, or y À3. Accordingly, A . 25. Let A aij and B bij be upper triangular matrices. Prove that AB is upper triangular with diagonal a11 b11 , a22 b22 ; . . ; ann bnn .