Suppose that P is a 4×5 matrix such that every solution of the equation Px = 0…
2021
Suppose that P is a 4×5 matrix such that every solution of the equation Px = 0 is a scalar multiple of [2 5 4 3 1]T. The rank of P is __________ .
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Correct answer: 4
Key idea: the nullspace (set of solutions to P x = 0) consists of all scalar multiples of the given vector, so the nullity is 1.
The matrix P has 5 columns, so the number of columns n = 5.
By the rank-nullity theorem (rank + nullity = number of columns),
rank = 5 - 1 = 4.
This is consistent with the fact that the rank cannot exceed the number of rows (4).
Answer: 4