# Linear Algebra: An Introduction, Second Edition by Richard Bronson

By Richard Bronson

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Additional info for Linear Algebra: An Introduction, Second Edition

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5 xn 2 3 b1 6b 7 6 27 7 b¼6 6 .. 7: 4 . 4 Linear Systems of Equations . 33 If m 6¼ n, then A is not square and the dimensions of x and b will be different. 20) is a set of scalar values for the variables x1 , x2 , . . , xn that when substituted into each equation of the system makes each equation true. Example 2 The scalar values x ¼ 2 and y ¼ 3 are a solution to the system 3x þ 2y ¼ 12 6x þ 4y ¼ 24 A second solution is x ¼ À4 and y ¼ 12. In contrast, the scalar values x ¼ 1, y ¼ 2, and z ¼ 3 are not a solution to the system 2x þ 3y þ 4z ¼ 20 4x þ 5y þ 6z ¼ 32 7x þ 8y þ 9z ¼ 40 because these values do not make the third equation true, even though they do satisfy the first two equations of the system.

34) A manufacturer produces desks and bookcases. Desks d require 5 hours of cutting time and 10 hours of assembling time. Bookcases b require 15 minutes of cutting time and 1 hour of assembling time. Each day the manufacturer has available 200 hours for cutting and 500 hours for assembling. How many desks and bookcases should be scheduled for completion each day to utilize all available workpower? (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 concept of reciprocals can be extended to matrices. The matrix counterpart of the number 1 is an identity matrix I, and the word inverse is used for a matrix A instead of reciprocal even though the notation AÀ1 is retained. Thus, a matrix B is an inverse of a matrix A if AB ¼ BA ¼ I in which case we write B ¼ AÀ1 . 5 The Inverse . 49 The requirement that a matrix commute with its inverse implies that both matrices are square and of the same order. Thus, inverses are only defined for square matrices.