By Richard Bronson

"Matrix equipment: utilized Linear Algebra, 3e, as a textbook, presents a special and finished stability among the idea and computation of matrices. the appliance of matrices isn't just for mathematicians. The use via different disciplines has grown dramatically through the years based on the quick alterations in expertise. Matrix tools is the essence of linear algebra and is what's used to aid physicalRead more...

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16. Find XD. 17. Find EF. 18. Find FE. 19. Find YF. 20. Find AB if A= 2 3 6 9 and B= 3 −1 −6 . 2 Note that AB = 0 but neither A nor B equals the zero matrix. 21. Find AB and CB if A= 3 1 2 , 0 2 1 4 , 2 2 4 x . y B= Thus show that AB = CB but A = C 22. Compute the product 1 3 C= 1 3 6 . 3 17 Matrix Multiplication 23. Compute the product ⎡ 1 ⎣3 1 0 1 3 ⎤⎡ ⎤ −1 x 1⎦ ⎣y⎦. 0 z 24. Compute the product a11 a21 a12 a22 x . y 25. Compute the product ⎡ b11 b21 b12 b22 ⎤ 2 ⎣−1⎦. 3 b13 b23 26. Evaluate the expression A2 − 4A − 5I for the matrix∗ A= 1 4 2 .

1. Determine (a) the probability that a brand X customer will be a brand X customer after 4 years, (b) after 6 years, and (c) the probability that a brand X customer will be a brand Y customer after 4 years. 40. A graph consists of a set of nodes, which we shall designate by positive integers, and a set of arcs that connect various pairs of nodes. 1. (b) Calculate M2 , and note that the i − j element of M2 is the number of paths consisting of two arcs that connect node i to node j. 2 41. 2. (b) Calculate M2 , and use that matrix to determine the number of paths consisting of two arcs that connect node 1 to node 5.

The reader who is already familiar with vectors will notice that we have not deﬁned vectors as directed line segments. We have done this intentionally, ﬁrst because in more than three dimensions this geometric interpretation loses its signiﬁcance, and second, because in the general mathematical framework, vectors are not directed line segments. However, the idea of representing a ﬁnite dimensional vector by its components and hence as a matrix is one that is acceptable to the scientist, engineer, and mathematician.