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Problem 8. Consider the 3 x 3 symmetric matrix over R 2 -2 A = 2 -2 -2 -2 6 (i) Let X be an mxn matrix. The column rank of X is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of X are equal (called the rank of X). Find the rank of A and denote it by k. (ii) Locate a k x k submatrix of A having rank k. (iii) Find 3 x 3 permutation matrices P and Q such that in the matrix PAQ the submatrix from (ii) is in the upper left portion of A.

Problem 8. Consider the 3 x 3 symmetric matrix over R 2 -2 A = 2 -2 -2 -2 6 (i) Let X be an mxn matrix. The column rank of X is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of X are equal (called the rank of X). Find the rank of A and denote it by k. (ii) Locate a k x k submatrix of A having rank k. (iii) Find 3 x 3 permutation matrices P and Q such that in the matrix PAQ the submatrix from (ii) is in the upper left portion of A.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
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Problem 8. Consider the 3 × 3 symmetric matrix over R 2 2 -2 A = 2 2 -2 -2 -2 6 (i) Let X be an mxn matrix. The column rank of X is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of X are equal (called the rank of X). Find the rank of A and denote it by k. (ii) Locate a k x k submatrix of A having rank k. (iii) Find 3 x 3 permutation matrices P and Q such that in the matrix PAQ the submatrix from (ii) is in the upper left portion of A.
Transcribed Image Text:Problem 8. Consider the 3 × 3 symmetric matrix over R 2 2 -2 A = 2 2 -2 -2 -2 6 (i) Let X be an mxn matrix. The column rank of X is the maximum number of linearly independent columns. The row rank is the maximum number of linearly independent rows. The row rank and the column rank of X are equal (called the rank of X). Find the rank of A and denote it by k. (ii) Locate a k x k submatrix of A having rank k. (iii) Find 3 x 3 permutation matrices P and Q such that in the matrix PAQ the submatrix from (ii) is in the upper left portion of A.
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