By Brown R.F. (ed.)

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Sample text

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.