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Chapter 3: Problem 22

Let \(A\) be an \(m \times n\) matrix. (a) Show that if \(B\) is a nonsingular \(m \times m\) matrix, then \(B A\) and \(A\) have the same null space and hence the same rank. (b) Show that if \(C\) is a nonsingular \(n \times n\) matrix, then \(A C\) and \(A\) have the same rank.

Short Answer

Expert verified
In summary, if B is a nonsingular m x m matrix, then BA and A have the same null space and rank. If C is a nonsingular n x n matrix, then AC and A have the same rank. This is proven by analyzing the implications of matrix multiplication on the null space and rank of a matrix, showing that the null spaces and column spaces are equal in both cases, resulting in equal ranks.

Step by step solution

01

Show BA and A have the same null space

Let x be a vector in the null space of A, such that Ax = 0. Since B is invertible, we have B^-1(BA)x = B^-1(0). Prduct of inverse of B and BA is given by (BA)x = 0. Thus, the null space of A is a subset of the null space of BA. Now, let x be a vector in the null space of BA, such that (BA)x = 0. Premultiply both sides of the equation by B^鈭�1 to obtain A(B^鈭�1(B)x) = A(0) which gives Ax=0. Therefore, the null space of BA is a subset of the null space of A. Thus, both null spaces are equal, and BA and A have the same null space.
02

Show that BA and A have the same rank

Since null spaces of BA and A are equal, the nullity of BA and A must be the same. By the rank-nullity theorem, we know that rank(A) + nullity(A) = n and rank(BA) + nullity(BA) = n, where n is the number of columns in matrix A. Since the nullity of BA and A is the same, the rank of BA and A must also be the same. (b) We want to show that if C is a nonsingular n x n matrix, then AC and A have the same rank.
03

Show that the column spaces of AC and A are equal

Let y be a vector in the column space of A; we can write y = Ax for some x. Define z = Cx, so that y = A(C^-1z) = (AC)(C^-1z). As C is nonsingular, C^-1 exists, and we can write (AC)(C^-1z) = ACw where w = C^-1z. Thus, y is in the column space of AC, and the column space of A is a subset of the column space of AC. Now let y be a vector in the column space of AC, we can write y = ACx for some x. Define z = Cx, so that y = A(C^-1z) = A(C^-1(Cw))=Aw as C^-1C=I. Thus, y is in the column space of A, and the column space of AC is a subset of the column space of A. Thus, both column spaces are equal, and A and AC have the same column space.
04

Show that AC and A have the same rank

Since A and AC have the same column space, their dimensions are the same, which implies that the rank of A and AC are equal. Therefore, if B is a nonsingular m x m matrix, BA and A have the same rank; If C is a nonsingular n x n matrix, AC and A have the same rank.

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