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Can i make conclusions about m x n matrix with rank r with (n-r) values and...

Can i make conclusions about m x n matrix with rank r with (n-r) values and it's solution number? Is there any other relations between rank of a matrix and avalaible solutions?

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Answer #1

Let A be a m x n matrix with rank r.

We presume that the solution number refer to the number of solutions to the equation AX = 0. The equation AX = 0 is always consistent as it always has the trivial solution. The homogenous system AX = 0 has non-trivial solutions if and only if there are free variables. If rank(A) = r, then there will be n-r free variables. The set of solutions to the equation AX = 0 is referred to as the null space of A and the dimension of the null space of A is called its nullity. Further, as per the dimension ( rank-nullity) theorem, the nullity of A = the number of columns in A- rank(A) = n-r. Thus the equation AX = 0 has n-r solutions. There is no other relationship between rank of a matrix and its nullity.

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