Therefore the area of the parallelogram determined by

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Therefore, the area of the parallelogram determined by and has to be times the area of the first one since one of the sides has been multiplied by this factor. Now, this last parallelogram, by Cavalieri's principle, has the same area as the parallelogram determined by and . Equating the areas of this last and the second parallelogram gives the equation Integer programming Ordinary differential equations Geometric interpretation
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from which Cramer's rule follows. This is a restatement of the proof above in abstract language. Consider the map , where is the matrix with substituted in the th column, as in Cramer's rule. Because of linearity of determinant in every column, this map is linear. Observe that it sends the th column of to the th basis vector (with 1 in the th place), because determinant of a matrix with a repeated column is 0. So we have a linear map which agrees with the inverse of on the column space; hence it agrees with on the span of the column space. Since is invertible, the column vectors span all of , so our map really is the inverse of . Cramer's rule follows. Geometric interpretation of Cramer's rule. The areas of the second and third shaded parallelograms are the same and the second is times the first. From this equality Cramer's rule follows. Other proofs A proof by abstract linear algebra
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A short proof of Cramer's rule [14] can be given by noticing that is the determinant of the matrix On the other hand, assuming that our original matrix A is invertible, this matrix has columns , where is the n -th column of the matrix A . Recall that the matrix has columns . Hence we have The proof for other is similar. Consider the system of three scalar equations in three unknown scalars and assign an orthonormal vector basis for as Let the vectors Adding the system of equations, it is seen that Using the exterior product, each unknown scalar can be solved as A short proof Proof using Clifford algebra
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For n equations in n unknowns, the solution for the k -th unknown generalizes to If a k are linearly independent, then the can be expressed in determinant form identical to Cramer ’s Rule as
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where ( c ) k denotes the substitution of vector a k with vector c in the k -th numerator position. A system of equations is said to be incompatible or inconsistent when there are no solutions and it is called indeterminate when there is more than one solution. For linear equations, an indeterminate system will have infinitely many solutions (if it is over an infinite field), since the solutions can be expressed in terms of one or more parameters that can take arbitrary values. Cramer's rule applies to the case where the coefficient determinant is nonzero. In the 2×2 case, if the coefficient determinant is zero, then the system is incompatible if the numerator determinants are nonzero, or indeterminate if the numerator determinants are zero. For 3×3 or higher systems, the only thing one can say when the coefficient determinant equals zero is that if any of the numerator determinants are nonzero, then the system must be incompatible. However, having all determinants zero does not imply that the system is indeterminate. A simple example where all determinants vanish (equal zero) but the system is still incompatible is the 3×3 system x+y+z=1, x+y+z=2, x+y+z=3.
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