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Unformatted text preview: Math 205, Fall 2006 B. Dodson Week 2: Ax = 0 , homog.; Matrix Inverse. The Rank of a matrix is defined to be the number of nonzero 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) uppertriagular matrix that is n × n with nonzero 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. (We’re using A # = ( A  b ) for the augmented matrix.) ———– Next, whenever rank( A ) = rank( A # ) we may readoff 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. 2 Finally, if r = rank( A ) = rank( A # ) < n, there are infinitely many solutions; and...
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 Spring '08
 zhang
 Math, Linear Algebra, Algebra, Matrices, Diagonal matrix, ej, B. Dodson

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