Notice there\'s two part of question, please expalin step by step Let A be 4 Tim

Question

Notice there's two part of question, please expalin step by step

Let A be 4 Times 3 matrix. Explain why the last row of rref(A) must be a zero row. Let A be a tall matrix (number of rows is greater than the number of columns). Explain why rref(A) must have a least one zero row. Theorem: Any matrix is reducible to just one reduced echelon form. Theorem: If A is an mxn matrix and n > m, then the homogeneous system Ax = 0 has infinitely many solutions. Theorem: If A is a square matrix and the reduced echelon of A is the identity matrix, then the homogeneous system Ax = 0 has only the zero solution. Theorem: If A is a square matrix and the homogeneous system Ax = 0 has only the zero solution, then the reduced echelon form of A is the identity matrix. Theorem: A linear system is consistent if and only the reduced echelon form of its augmented matrix has no row of the form [0 ... 0 |1].

Explanation / Answer

a) The rank is less than 3 ( number of columns) so rref(A) must have one zero row (see note below )

b) The rank will be smaller or equal to the number of columns. So since we have more rows than columns, then in the row echelon form matrix we will have more than one zero row.

( Remember that the rank is the number of non-zero rows in rref(A) )

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