Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A) rank(B) = b, and a > b. = a, a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and to the null space of B. b) Based on a), which of the following statements is true? a. rank (A+B) = a+b b. rank(A+B) > a c. rank (A+B) < a c) What is the rank of the matrix A²
Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A) rank(B) = b, and a > b. = a, a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and to the null space of B. b) Based on a), which of the following statements is true? a. rank (A+B) = a+b b. rank(A+B) > a c. rank (A+B) < a c) What is the rank of the matrix A²
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.6: Rank Of A Matrix And Systems Of Linear Equations
Problem 69E: Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of...
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![Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A)
rank(B) = b, and a > b.
= a,
a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and
to the null space of B.
b) Based on a), which of the following statements is true?
a. rank (A+B) = a+b
b. rank(A+B) > a
c. rank (A+B) < a
c) What is the rank of the matrix A²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2Fa4b2d2d8-d86c-45f2-b450-e1af91f3ad21%2Fcs4pykp_processed.png&w=3840&q=75)
Transcribed Image Text:Consider two positive semi-definite N × N-dimensional matrices A and B, with rank(A)
rank(B) = b, and a > b.
= a,
a) Prove that all vectors belonging to the null space of A + B belong to the null space of A and
to the null space of B.
b) Based on a), which of the following statements is true?
a. rank (A+B) = a+b
b. rank(A+B) > a
c. rank (A+B) < a
c) What is the rank of the matrix A²
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