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Question 381808: let a be an nxn matrix. let Ax=b be the matrix equation for a linear system in which b equals k times the ith column of A, where k is some non-zero constant. Use cramer's rule to prove that xi=k and xj=0 for al j does not equal i is the single unique solution to the system.
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Assume |A| is not equal to 0. I will denote by |A(n)| the determinant of the matrix with the nth column replaced by b. Then xn = |A(n)|/|A|, by Cramer's rule. Now for n not equal to i, |A(n)| = 0, because the nth column of matrix A(n) would be k times the ith column of A(n), and by property of determinants,
|A(n)| = 0, and so xn = |A(n)|/|A| = 0 as long as n is not equal to i.
If n = i, then |A(i)| = k|A|, and so xi = |A(i)|/|A|= k, and the solution is complete.
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