Unformatted text preview: combining it with the others results in 20. Suppose that A is nonsingular, and that there exist matrices B and C, such that
AB I and AC I Based on the properties of matrices, it follows that
AB C AY 0 Write the difference of the two matrices, Y , in terms of its columns as
Y yy y. The th column of the product matrix, AY , can be expressed as A yj . Now since all
columns of the product matrix consist only of zeros, we end up with homogeneous
systems of linear equations
A yj 0 , Since A is nonsingular, each system must have a trivial solution. That is, yj
for
. Hence Y 0
and B C
21 0 , . ________________________________________________________________________
page 349 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” CHAPTER 7. â€”â€” A B . Based on the standard definition of matrix multiplication,
AB .
A . Note that
A C. Therefore
A .
The result can also be written as 23. First note that ________________________________________________________________________
page 350 â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€...
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 Spring '08
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 Linear Algebra, eigenvector

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