Consistent State: A System of equations having one or more solutions is called a consistent system of equations.There are two states of the Linear equation system: Here P is rank of gauss elimination representation of AX = B. If P = P(A) ≠ number of unknown, infinite number of solutions.If P = P(A) = the number of unknown variables, unique solution.System of non-homogeneous linear equations AX = B. If P(A) If P(A) = number of unknowns, unique solution.is always a solution means all the unknowns has same value as zero. System of homogeneous linear equations AX = 0. The rank of a skew symmetric matrix cannot be equal to one.If A m×1 is a non zero column matrix and B 1×n is a non zero row matrix then P(AB) = 1.If A and B are square matrices of order n then P(AB) ? P(A) + P(B) – n.If P(A) = m and P(B)=n then P(AB) ≤ min(m,n).Rank of a matrix A mxn, P(A) ≤ min(m,n).If I n is the nxn unit matrix then P(A) = n.If A is a null matrix then P(A) = 0 i.e.If A is a non-singular matrix of order n, then rank of A = n i.e. All the determinants of square sub-matrices of order (r+1) or higher than r are zero.It has at least one square sub-matrices of order r who has non-zero determinant.A matrix is said to be of rank r, if it satisfies the following properties: Let A be any mxn matrix and it has square sub-matrices of different orders. Rank of a matrix: Rank of matrix is the number of non-zero rows in the row reduced form or the maximum number of independent rows or the maximum number of independent columns. Mathematics | Rings, Integral domains and Fields.Mathematics | Independent Sets, Covering and Matching.Mathematics | Sequence, Series and Summations.Mathematics | Generating Functions – Set 2.Discrete Maths | Generating Functions-Introduction and Prerequisites.Mathematics | Total number of possible functions.Mathematics | Classes (Injective, surjective, Bijective) of Functions.Number of possible Equivalence Relations on a finite set.Mathematics | Closure of Relations and Equivalence Relations.Mathematics | Representations of Matrices and Graphs in Relations.Discrete Mathematics | Representing Relations.Mathematics | Introduction and types of Relations.
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