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In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the ...
Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix N that is its inverse (these are equivalent under Cramer's rule). Thus every equation Mx = b, where M and b both have integer components and M is unimodular, has an integer solution.
Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns. Cramer's rule is useful for reasoning about the solution, but, except for n = 2 or 3 , it is rarely used for computing a solution, since Gaussian elimination is a faster algorithm.
The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings. The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group, the general linear group of degree n, denoted GL n (R).
Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. (Indeed, large determinants are most easily computed using row reduction.)
Miller's rule, in optics, is an empirical rule which gives an estimate of the order of magnitude of the nonlinear coefficient. Monro-Kellie doctrine: The pressure–volume relationship between intracranial contents and cerebral perfusion pressure (CPP) states that the cranial compartment is inelastic and that the volume inside the cranium is ...
Cramer's rule (See [2] or section 4.1. [3]) The inverse to a Manin matrix M can be defined by the standard formula: M − 1 = 1 det c o l ( M ) M a d j , {\displaystyle M^{-1}={\frac {1}{{\det }^{col}(M)}}M^{adj},} where M adj is adjugate matrix given by the standard formula - its (i,j)-th element is the column-determinant of the (n − 1) × ...
A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I]