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kwal0203
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Homework Statement
a.) If A is an 'n x n' matrix and X is an 'n x 1' nonzero column matrix with
AX = 0
show, by assuming the contrary, that det(A) = 0
b.) Using the answer in 'a' show that the scalar equation which gives the values of λ that satisfy the matrix equation AX = λIX is:
det(A - λI) = 0
2. The attempt at a solution
a.) If det(A) ≠ 0 then A^-1 exists.
X = A^-1 x (AX) = A^-1 x (0) = 0
This is a contradiction because x nonzero so det(A) = 0... this bit I understand however the next part
b.) AX = λIX -> X(A - λI) = 0
X ≠ 0 so for the equation to be true A - λI = 0
I'm not sure how to apply the first result to the second question? It's from Bimore & Davies Calculus: Concepts and Methods book.
Any help would be appreciated!