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Proof: Let be a monotone matrix and assume there exists with . Then, by monotonicity, and , and hence .
Let be a real square matrix. is monotone if and only if .[1]
Proof: Suppose is monotone. Denote by the -th column of . Then, is the -th standard basis vector, and hence by monotonicity. For the reverse direction, suppose admits an inverse such that . Then, if , , and hence is monotone.
Examples
The matrix is monotone, with inverse .
In fact, this matrix is an M-matrix (i.e., a monotone L-matrix).
Note, however, that not all monotone matrices are M-matrices. An example is , whose inverse is .
^ abcMangasarian, O. L. (1968). "Characterizations of Real Matrices of Monotone Kind". SIAM Review. 10 (4): 439–441. doi:10.1137/1010095. ISSN0036-1445.