**Determinant** ([edit](http://github.com/nicebyte/wiki/edit/master/Wiki/Determinant.md)) # Definition A determinant can be thought of as an operation on $n$ $n$-dimensional vectors. Given the vectors $\overline{a}_1 = (a_{11}, a_{12}, ... a_{1n})$, $\overline{a}_2 = (a_{20}, a_{21}, ... a_{2n})$, $...$, $\overline{a}_n = (a_{n1}, a_{n2}, ..., a_{nn})$, we define the determinant as follows: $$ \det \boldsymbol{A} = \sum_{\sigma \in S_n} (sgn(\sigma)\prod_{i=1}^n a_{i \sigma_i}) $$ where $\boldsymbol{A}$ is the Matrix whose columns are the vectors $\overline{a}_1, \overline{a}_2, ... \overline{a}_n$, and $S_n$ is the set of Permutations over $n$ (Leibniz formula). A determinant can also be thought of as a property of a matrix, however only square matrices have that property. The definition is rather unwieldy, and in practice it might be easier to remember the recursive method for computing the determinant (see below). # Examples For two two-dimensional vectors: $$ \det \begin{bmatrix}a && c \\ b && d \end{bmatrix} = ad - cb $$ For three three-dimensional vectors: $$ \det \begin{bmatrix} a && d && g \\ b && e && h \\ c && f && i \end{bmatrix} = a(ei-hf)-d(bi-he)+g(bf-ec) $$ # Computing the Determinant Given a $n \times n$ matrix $ \boldsymbol{A} $, we define $ \boldsymbol{A}_{ij} $ as the determinant of the matrix obtained from $ \boldsymbol{A} $ by removing the $i$-th column and $j$-th row. The determinant can be computed as follows: $$ \det \boldsymbol{A} = \sum_{i=1}^n (-1)^{k+i} a_{ki} \boldsymbol{A}_{ki} = \sum_{i=1}^n (-1)^{i+p} a_{ip} \boldsymbol{A}_{ip} $$ where $k$ and $p$ are an arbitrary column or row, respectively. # Connection to Eigenvalues If $\lambda_1, \lambda_2, ... \lambda_n$ are the Eigenvalues of the matrix $\boldsymbol{A}$, then $\det \boldsymbol{A} = \lambda_1 \lambda_2 ... \lambda_n$. Since eigenvalues represent how much the transform corresponding to $\boldsymbol{A}$ stretches or compresses the space in corresponding directions, $\det \boldsymbol{A} = 1$ implies that the transform preserves volume. Categories: Mathematics, LinearAlgebra