Question

5. Find a matrix A with the following properties or explain why it cannot exist: (a) A is a 3 x 4 matrix with rank 2 and 3 Null(A) = span { (b) A is a 4 x 4 matrix with nullity 2 and 3 Col(A) = span 5 (c) A is a 3 x 4 matrix with nullity 2 and 3 Col(A) = span 5

Answer 1

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a) matrix A is 3*4 ..

its rank = 2 ..

its nullity = 4 - rank(A) = 4 - 2 = 2  

so , as its nullity is 2 .. therefore null space contains 2 elemnts but here we have only one element in null space

so , not possible ..

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b)

nullity = 2

col (A) have 3 elements but 2nd element is 1st - 3rd .. so bascillay dim of col(A) is 2 . so matrix possible ..

3 6 A= 3 3 0 0 4 5 -1 6 12 -2 -2 0 0

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c)  

nullity = 2 and dim of col(A) is 3 .. not possible because nullity + dim of col(A) must be 4 ..

so , matrix not possible