By Brown R.F. (ed.)

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27 28 Linear Algebra (9) Use the matrices defined in Problems (7) and (8), partitioned into block diagonal form, to calculate AB. (10) Use partitioning to calculate A2 and A3 for 2 1 60 6 60 A¼6 60 6 40 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 3 0 07 7 07 7: 07 7 15 0 What is An for any positive integer n > 3? (11) Determine which, if any, of the following matrices are in row-reduced form: 2 3 2 3 0 1 0 4 À7 1 1 0 4 À7 60 0 0 1 6 27 27 7, B ¼ 6 0 1 0 1 7, A¼6 40 0 0 0 5 4 1 0 0 1 0 15 0 0 0 0 0 0 0 0 1 5 2 1 60 6 C¼4 0 0 1 1 0 0 0 0 0 0 3 À7 27 7, 15 5 4 1 0 1 2 2 2 0 3 2 2 5, 2 2 0 0 3 3 1 5, 0 2 E ¼ 40 0 2 1 G ¼ 40 1 2 0 J ¼ 41 0 2 2 L ¼ 40 0 2 1 N ¼ 40 0 1 0 0 3 1 2 5, 0 3 0 0 2 0 5, 0 0 0 0 0 3 0 1 5, 0 2 0 60 D¼6 40 0 2 0 F ¼ 40 0 2 0 H ¼ 40 0 2 1 K ¼ 40 0 2 1 0 0 0 0 0 1 0 0 0 0 3 0 0 5, 0 0 1 0 3 0 0 5, 0 0 À1 0 3 2 1 5, 0 1 M ¼ 40 0 1=2 1 0 0 1 !

35) A mining company has a contract to supply 70,000 tons of low-grade ore, 181,000 tons of medium-grade ore, and 41,000 tons of high-grade ore to a supplier. The company has three mines that it can work. Mine A produces 8000 tons of low-grade ore, 5000 tons of medium-grade ore, and 1000 tons of high-grade ore during each day of operation. Mine B produces 3000 tons of low-grade ore, 12,000 tons of medium-grade ore, and 3000 tons of high-grade ore for each day it is in operation. The figures for mine C are 1000, 10,000, and 2000, respectively.

5 6 1 2 are both submatrices of A. In contrast, and |1 2| are not 8 9 8 9 ! 1 2 minors because is not a submatrix of A and [1 2], although a submatrix 8 9 of A, is not square. then 1 7 ! 2 8 ! CHAPTER 1 and 5 8 6 9 ! If A ¼ [aij] is a square matrix, then the cofactor of the element aij is the scalar obtained by multiplying (À1)iþj with the minor obtained from A by removing the jth row and jth column. In other words, to compute the cofactor of an element aij in a matrix A, first form a submatrix of A by deleting from A both the row and column in which the element aij appears, then calculate the determinant of that submatrix, and finally multiply the determinant by the number (À1)iþj.