Editing Determinant (section)

=== Cramer's rule ===

Determinants can be used to describe the solutions of a [[linear system of equations]], written in matrix form as <math>Ax = b</math>. This equation has a unique solution <math>x</math> if and only if <math>\det (A)</math> is nonzero. In this case, the solution is given by [[Cramer's rule]]:
:<math>x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, 2, 3, \ldots, n</math>

where <math>A_i</math> is the matrix formed by replacing the <math>i</math>-th column of <math>A</math> by the column vector <math>b</math>. This follows immediately by column expansion of the determinant, i.e.
:<math>\det(A_i) =
  \det\begin{bmatrix}a_1 & \ldots & b & \ldots & a_n\end{bmatrix} 
</math>
<math>
  =\sum_{j=1}^n x_j\det\begin{bmatrix}a_1 & \ldots & a_{i-1} & a_j & a_{i+1} & \ldots & a_n\end{bmatrix} =
  x_i\det(A)
</math>

where the vectors <math>a_j</math> are the columns of ''A''.  The rule is also implied by the identity

:<math>A\, \operatorname{adj}(A) = \operatorname{adj}(A)\, A = \det(A)\, I_n.</math>

Cramer's rule can be implemented in <math>\operatorname O(n^3)</math> time, which is comparable to more common methods of solving systems of linear equations, such as [[LU decomposition|LU]], [[QR decomposition|QR]], or [[singular value decomposition]].<ref>{{harvnb|Habgood|Arel|2012}}</ref>