- #1
Tim 1234
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Homework Statement
Suppose A is non-singular (nxn) matrix. Given that AB=(zero matrix), show that B=(zero matrix).
Hint: Express A as [A1, A2, ..., An]
and Express AB as [AB1, AB2, ..., ABn]
Homework Equations
The Attempt at a Solution
[/B]
I argued that because A is non-singular, A=[A1, A2, ..., An] is linearly independent, thus Ax=(theta) has the unique solution x=(theta).
That is, for x1A1+x2A2+...+xnAn=(theta), x1=x2=...=xn=0.
Further, if A is a zero matrix, Ax=(theta) cannot have the unique solution x=(theta).
Because A cannot be a zero matrix and AB=(zero matrix), B=(zero matrix) by necessity.
Is my reasoning correct in that a non-singular matrix cannot be a zero matrix?
Thanks