Unformatted text preview: 1 (5) into the original equation
? 3x = 5
3 3−1 (5) ? =5
? 3 · 3−1 (5) = 5 Associative property of multiplication
? 1·5 = 5 Inverse property 5=5 Multiplicative Identity Thinking back to Theorem 8.5, we know that matrix multiplication enjoys both an associative
property and a multiplicative identity. What’s missing from the mix is a multiplicative inverse for
the coeﬃcient matrix A. Assuming we can ﬁnd such a beast, we can mimic our solution (and check)
to 3x = 5 as follows
1 Every nonzero real number a has a multiplicative inverse, denoted a−1 , such that a−1 · a = a · a−1 = 1. 494 Systems of Equations and Matrices Solving AX = B Checking our answer
? AX
A−1 (AX )
A−1 A X
I2 X
X =
=
=
=
= AX = B B
A−1 B
A−1 B
A−1 B
A−1 B A A−1 B ? =B
? AA−1 B = B
? I2 B = B
B=B The matrix A−1 is read ‘Ainverse’ and we will deﬁne it formally later in the section. At this stage,
we have no idea if such a matrix A−1 exists, but that won’t deter us from trying to ﬁnd it.2 We
want A−1 to satisfy two equations, A−1 A = I2 and AA−1 = I2 , maki...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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