Suppose P is a 4 x 5 matrix such that every solution of Px = 0 is a scalar multiple of [2, 5, 4, 3,

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Question

Suppose P is a 4 x 5 matrix such that every solution of Px = 0 is a scalar multiple of [2, 5, 4, 3, 1]^T. The rank of P is ___

Accepted Answer

Answer: 4Concept -- Rank-Nullity Theorem (from Gilbert Strang, Linear Algebra and Its Applications):The Rank-Nullity Theorem states: rank(P) + nullity(P) = number of columns of P.P is a 4 x 5 matrix (4 rows, 5 columns). The solution set of Px = 0 is the null space of P.The null space is spanned by a single vector [2,5,4,3,1]^T (every solution is a scalar multiple). This means the null space has dimension 1 (nullity = 1).By the Rank-Nullity Theorem: rank(P) = 5 - nullity(P) = 5 - 1 = 4.Verification: rank(P) <= min(rows, columns) = min(4,5) = 4. Since rank = 4 = number of rows, the matrix P has full row rank.

Suppose P is a 4 x 5 matrix such that every solution of Px = 0 is a scalar multiple of [2, 5, 4, 3, 1]^T. The rank of P is ___

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Suppose P is a 4 x 5 matrix such that every solution of Px = 0 is a scalar multiple of [2, 5, 4, 3, 1]^T. The rank of P is ___

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