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I have some more linear algebra problems...

First: Prove that if B is a 3x1 matrix and C is a 1x3 matrix, then the 3x3 matrix BC has rank at most 1. Conversely, show that if A is any 3x3 matrix having rank 1, then there exist a 3x1 matrix B and a 1x3 matrix C such that A=BC

The first part is easy (it follows from a theorem). Im not sure how to do the "Conversely" part, and im also curious about whether it generalizes to mxn matrices and what the linear transformation analogy to this would be.

Second: Let A be an mxn matrix with rank m. Prove that there exists an nxm matrix B s.t AB = I

Also, let B be an nxm matrix with rank m. Prove that there exists an mxn matrix A such that AB = I

Im not sure about these. I know that since the rank is m, there are m linearly independent rows...

Any help would be useful. Thanks.

First: Prove that if B is a 3x1 matrix and C is a 1x3 matrix, then the 3x3 matrix BC has rank at most 1. Conversely, show that if A is any 3x3 matrix having rank 1, then there exist a 3x1 matrix B and a 1x3 matrix C such that A=BC

The first part is easy (it follows from a theorem). Im not sure how to do the "Conversely" part, and im also curious about whether it generalizes to mxn matrices and what the linear transformation analogy to this would be.

Second: Let A be an mxn matrix with rank m. Prove that there exists an nxm matrix B s.t AB = I

Also, let B be an nxm matrix with rank m. Prove that there exists an mxn matrix A such that AB = I

Im not sure about these. I know that since the rank is m, there are m linearly independent rows...

Any help would be useful. Thanks.

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