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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.
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.)
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.
Cramer's rule: In linear algebra, 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. Named after Swiss mathematician Gabriel Cramer .
a foreshortening technique using N scale model trains in the background (distance) with HO scale in the foreground. mixing 1:43 scale, 1:48 scale and 1:50 scale die-cast models with O scale model trains. using Matchbox cars (1:64 to 1:100) with HO scale and S scale. mixing OO scale British model trains with HO scale models. Both scales run on ...
An example of a degenerate case, in which n(n + 3) / 2 points on the curve are not sufficient to determine the curve uniquely, was provided by Cramer as part of Cramer's paradox. Let the degree be n = 3, and let nine points be all combinations of x = −1, 0, 1 and y = −1, 0, 1.
When H0 scale was being introduced, the motors available were too large [4] to fit in scale-sized bodies and so as a compromise the scale was increased from 3.5 mm to 4 mm to the foot, but the gauge was not changed so other elements could be shared. For 00 therefore the track is about 12.5% narrower than it should be for the scale used.