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Question: 3 true or false if a homogeneous system of two...

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3. True or false: If a homogeneous system of two linear equations in two variables has exactly one solution, then the system of equations obtained by taking the transpose of the coefficient matrix also has exactly one solution.

In other words, if the system

ax + by = 0

cx + dy = 0

has exactly one solution, then the system

ax + cy = 0

bx + dy = 0

also has exactly one solution.

(Give a proof or counterexample.) (Explain)

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