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Unformatted text preview: Math 205, Spring 2006 B. Dodson Week 2: Ax = 0 , homog.; Matrix Inverse. The Rank of a matrix is defined to be the number of non-zero rows in the RREF of the matrix. We do not necessarily need the RREF or even a REF to find the rank; for example, a (square) upper-triagular matrix that is n n with non-zero diagonal entries always has rank n. (why?) We will discuss the results of the qualitative theory more later on the text; but the first main result is that when Ax = b, with A an m n matrix, has rank( A ) = rank( A # ) = n, the system has a unique solution. (Were using A # = ( A | b ) for the augmented matrix.) 2 Next, whenever rank( A ) = rank( A # ) we may read-off a particular solution x p , so that the system is consistent; while rank( A ) = rank( A # ) occurs only when rank( A # ) = rank( A ) +1, in which case the last equation reads 0 = 1 , which is inconsistent, so the system has no solution....
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