Editing Determinant (section)

=== Laplace expansion ===
[[Laplace expansion]] expresses the determinant of a matrix <math>A</math> [[Recursion|recursively]] in terms of determinants of smaller matrices, known as its [[minor (matrix)|minors]]. The minor <math>M_{i,j}</math> is defined to be the determinant of the <math>(n-1) \times (n-1)</math> matrix that results from <math>A</math> by removing the <math>i</math>-th row and the <math>j</math>-th column. The expression <math>(-1)^{i+j}M_{i,j}</math> is known as a [[cofactor (linear algebra)|cofactor]]. For every <math>i</math>, one has the equality
:<math>\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j},</math>
which is called the ''Laplace expansion along the {{mvar|i}}th row''. For example, the Laplace expansion along the first row (<math>i=1</math>) gives the following formula:
:<math>
  \begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix} =
  a\begin{vmatrix}e&f\\ h&i\end{vmatrix} - b\begin{vmatrix}d&f\\ g&i\end{vmatrix} + c\begin{vmatrix}d&e\\ g&h\end{vmatrix}
</math>
Unwinding the determinants of these <math>2 \times 2</math>-matrices gives back the Leibniz formula mentioned above. Similarly, the ''Laplace expansion along the <math>j</math>-th column'' is the equality
:<math>\det(A)= \sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}.</math>
Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the [[Vandermonde matrix]]
<math display="block">\begin{vmatrix}
            1 &         1 &         1 & \cdots &         1 \\
          x_1 &       x_2 &       x_3 & \cdots &       x_n \\
        x_1^2 &     x_2^2 &     x_3^2 & \cdots &     x_n^2 \\
       \vdots &    \vdots &    \vdots & \ddots &    \vdots \\
    x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1}
  \end{vmatrix} =
  \prod_{1 \leq i < j \leq n} \left(x_j - x_i\right).
</math>The ''n''-term Laplace expansion along a row or column can be [[Laplace expansion#Laplace expansion of a determinant by complementary minors|generalized]] to write an ''n'' x ''n'' determinant as a sum of <math>\tbinom nk</math> [[Binomial coefficient|terms]], each the product of the determinant of a ''k'' x ''k'' [[Minor (linear algebra)|submatrix]] and the determinant of the complementary (''n−k'') x (''n−k'') submatrix.