- #1
bernoli123
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check that, for any nxn matrices A,B then rank(AB) (> or =) rank A +rank(B)-n
The rank of a matrix is the maximum number of linearly independent rows or columns in that matrix. In other words, it is the number of rows or columns that contribute to the linearly independent combinations of the matrix.
The nxn matrices A and B determine the rank of AB because the rank of AB is equal to the minimum of the ranks of A and B. In other words, the rank of AB can be no greater than the smaller rank of A or B.
The rank of AB is directly related to the number of solutions of linear systems. If the rank of AB is equal to the number of unknowns in the system, then there is a unique solution. If the rank is less than the number of unknowns, then there are either infinitely many solutions or no solution at all.
No, the rank of AB can never be greater than the smaller rank of A or B. This is because the rank of AB is determined by the linearly independent combinations of both A and B, so it cannot be greater than the number of linearly independent combinations in either A or B.
The rank of a matrix A can be determined by performing row operations on the matrix and counting the number of non-zero rows in the resulting matrix. This is also known as the row-reduced echelon form of the matrix. The number of non-zero rows will be the rank of A.