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winter 2007assign3

# winter 2007assign3 - or null space of A(b Prove that r A<...

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Math 235 Assignment 3 Due 9:15am, Wednesday Jan 31, 2007. 1. From the Text § 5.1 #16, #22 #24, #26, and #32. § 5.2#14. 2. If A is an m × n matrix, prove that the r ( A ) (the rank of A ) is equal to the largest integer k for which there is a k × k submatrix of A that is non-singular. 3. Let A be an n × n matrix such that each row sum and each column sum is equal to 0 . (a) Let 1 = (1 , . . . , 1) , with n components. Prove that 1 T ker ( A ) (where ker A is the kernel
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Unformatted text preview: or null space of A ). (b) Prove that r ( A ) < n . (c) If r ( A ) < n-1 , prove that cof i,j ( A ) = 0 for all 1 ≤ i < j ≤ n, where cof i,j A denotes the ( i, j )-cofactor of A . (d) Prove that ( cof i, 1 ( A ) , . . . , cof i,n ( A )) T ∈ ker A , for i = 1 , . . . , n. (e) Hence, or otherwise, prove that the value of cof i,j is independent of i and j....
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