Stitz-Zeager_College_Algebra_e-book

Det an detan for all natural numbers n a is invertible

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Unformatted text preview: tion to serve us in solving systems of linear equations. To that end, we begin by defining the product of a row and a column. We motivate the general definition with an example. Consider the two matrices A and B below. 3 1 2 −8 2 0 −1 8 −5 9 B= 4 A= −10 3 5 5 0 −2 −12 Let R1 denote the first row of A and C 1 denote the first column of B . To find the ‘product’ of R1 with C 1, denoted R1 · C 1, we first find the product of the first entry in R1 and the first entry in C 1. Next, we add to that the product of the second entry in R1 and the second entry in C 1. Finally, we take that sum and we add to that the product of the last entry in R1 and the last entry in C 1. Using entry notation, R1 · C 1 = a11 b11 + a12 b21 + a13 b31 = (2)(3)+(0)(4)+(−1)(5) = 6+0+(−5) = 1. We can visualize this schematically as follows 3 1 2 −8 2 0 −1 4 8 −5 9 −10 3 5 5 0 −2 −12 3 3 3 −− − − − − − − −→ −− − − − − − − −→ −− − − − − −...
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