Characterization of the determinant The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an n \times n matrix
A as being composed of its n columns, so denoted as :A = \big ( a_1, \dots, a_n \big ), where the
column vector a_i (for each
i) is composed of the entries of the matrix in the
i-th column. • \det\left(I\right) = 1, where I is an
identity matrix. • The determinant is
multilinear: if the
jth column of a matrix A is written as a
linear combination a_j = r \cdot v + w of two
column vectors
v and
w and a number
r, then the determinant of
A is expressible as a similar linear combination: • : \begin{align}|A| &= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\ &= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n | \end{align} • The determinant is
alternating: whenever two columns of a matrix are identical, its determinant is 0: • : | a_1, \dots, v, \dots, v, \dots, a_n| = 0. If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any n \times n matrix
A a number that satisfies these three properties. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula. To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a
standard basis vector. These determinants are either 0 (if the columns are linearly dependent, by property 3) or else ±1 (by property 1 and 3 - the minus sign appears when the columns are permuted according to an
odd permutation), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.
Immediate consequences These rules have several further consequences: • The determinant is a
homogeneous function, i.e., \det(cA) = c^n\det(A) (for an n \times n matrix A). • Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): |a_1, \dots, a_j, \dots a_i, \dots, a_n| = - |a_1, \dots, a_i, \dots, a_j, \dots, a_n|. This formula can be applied iteratively when several columns are swapped. For example |a_3, a_1, a_2, a_4 \dots, a_n| = - |a_1, a_3, a_2, a_4, \dots, a_n| = |a_1, a_2, a_3, a_4, \dots, a_n|. Yet more generally, any permutation of the columns multiplies the determinant by the
sign of the permutation. • If some column can be expressed as a linear combination of the
other columns (i.e. the columns of the matrix form a
linearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0. • Adding a scalar multiple of one column to
another column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating. • If A is a
triangular matrix, i.e. a_{ij}=0, whenever i>j or, alternatively, whenever i, then its determinant equals the product of the diagonal entries: \det(A) = a_{11} a_{22} \cdots a_{nn} = \prod_{i=1}^n a_{ii}. Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a
diagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation \sigma which gives a non-zero contribution is the identity permutation.
Example These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact,
Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix A using that method: :A = \begin{bmatrix} -2 & -1 & 2 \\ 2 & 1 & 4 \\ -3 & 3 & -1 \end{bmatrix}. Combining these equalities gives |A| = -|E| = -(18 \cdot 3 \cdot (-1)) = 54.
Transpose The determinant of the
transpose of A equals the determinant of
A: :\det\left(A^\textsf{T}\right) = \det(A). This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an matrix as being composed of
n rows, the determinant is an
n-linear function.
Multiplicativity and matrix groups The determinant is a
multiplicative map, i.e., for square matrices A and B of equal size, the determinant of a
matrix product equals the product of their determinants: :\det(AB) = \det (A) \det (B) This key fact can be proven by observing that, for a fixed matrix B, both sides of the equation are alternating and multilinear as a function depending on the columns of A. Moreover, they both take the value \det B when A is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim. A matrix A with entries in a
field is
invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by :\det\left(A^{-1}\right) = \frac{1}{\det(A)} = [\det(A)]^{-1}. In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size n over a field K) forms a group known as the
general linear group \operatorname{GL}_n(K) (respectively, a
subgroup called the
special linear group \operatorname{SL}_n(K) \subset \operatorname{GL}_n(K). More generally, the word "special" indicates the subgroup of another
matrix group of matrices of determinant one. Examples include the
special orthogonal group (which if
n is 2 or 3 consists of all
rotation matrices), and the
special unitary group. Because the determinant respects multiplication and inverses, it is in fact a
group homomorphism from \operatorname{GL}_n(K) into the multiplicative group K^\times of nonzero elements of K. This homomorphism is surjective and its kernel is \operatorname{SL}_n(K) (the matrices with determinant one). Hence, by the
first isomorphism theorem, this shows that \operatorname{SL}_n(K) is a
normal subgroup of \operatorname{GL}_n(K), and that the
quotient group \operatorname{GL}_n(K)/\operatorname{SL}_n(K) is isomorphic to K^\times. The
Cauchy–Binet formula is a generalization of that product formula for
rectangular matrices. This formula can also be recast as a multiplicative formula for
compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.
Laplace expansion Laplace expansion expresses the determinant of a matrix A
recursively in terms of determinants of smaller matrices, known as its
minors. The minor M_{i,j} is defined to be the determinant of the (n-1) \times (n-1) matrix that results from A by removing the i-th row and the j-th column. The expression (-1)^{i+j}M_{i,j} is known as a
cofactor. For every i, one has the equality :\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j}, which is called the
Laplace expansion along the th row. For example, the Laplace expansion along the first row (i=1) gives the following formula: : \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} Unwinding the determinants of these 2 \times 2-matrices gives back the Leibniz formula mentioned above. Similarly, the
Laplace expansion along the j-th column is the equality :\det(A)= \sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}. 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 \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 The
n-term Laplace expansion along a row or column can be
generalized to write an
n x
n determinant as a sum of \tbinom nk
terms, each the product of the determinant of a
k x
k submatrix and the determinant of the complementary (
n−k) x (
n−k) submatrix.
Adjugate matrix The
adjugate matrix \operatorname{adj}(A) is the transpose of the matrix of the cofactors, that is, : (\operatorname{adj}(A))_{i,j} = (-1)^{i+j} M_{ji}. For every matrix, one has : (\det A) I = A\operatorname{adj}A = (\operatorname{adj}A)\,A. Thus the adjugate matrix can be used for expressing the inverse of a
nonsingular matrix: : A^{-1} = \frac 1{\det A}\operatorname{adj}A.
Block matrices The formula for the determinant of a 2 \times 2 matrix above continues to hold, under appropriate further assumptions, for a
block matrix, i.e., a matrix composed of four submatrices A, B, C, D of dimension m \times m, m \times n, n \times m and n \times n, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the
Schur complement, is :\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det(A) \det(D) = \det\begin{pmatrix}A& B\\ 0& D\end{pmatrix}. If A is
invertible, then it follows with results from the section on multiplicativity that :\begin{align} \det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(A)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}A^{-1}& -A^{-1} B\\ 0& I_n\end{pmatrix}}_{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\ & = \det(A) \det\begin{pmatrix}I_m& 0\\ C A^{-1}& D-C A^{-1} B\end{pmatrix}\\ & = \det(A) \det(D - C A^{-1} B), \end{align} which simplifies to \det (A) (D - C A^{-1} B) when D is a 1 \times 1 matrix. A similar result holds when D is invertible, namely :\begin{align} \det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(D)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}I_m& 0\\ -D^{-1} C& D^{-1}\end{pmatrix}}_{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\ & = \det(D) \det\begin{pmatrix}A - B D^{-1} C& B D^{-1}\\ 0& I_n\end{pmatrix}\\ & = \det(D) \det(A - B D^{-1} C). \end{align} Both results can be combined to derive
Sylvester's determinant theorem, which is also stated below. If the blocks are square matrices of the
same size further formulas hold. For example, if C and D
commute (i.e., CD=DC), then :\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(AD - BC). This formula has been generalized to matrices composed of more than 2 \times 2 blocks, again under appropriate commutativity conditions among the individual blocks. For A = D and B = C, the following formula holds (even if A and B do not commute). :\det\begin{pmatrix}A & B\\ B & A\end{pmatrix} = \det\begin{pmatrix}A+B & B\\ B+A & A\end{pmatrix} = \det\begin{pmatrix}A+B & B\\ 0 & A-B\end{pmatrix} = \det(A+B) \det(A-B). It is possible to compute the determinant by the block matrices in a fast way with the use of
fast matrix multiplication algorithms in the time O({n^\omega }) for ~2.37 \le \omega , by the LU decomposition.
Sylvester's determinant theorem Sylvester's determinant theorem states that for
A, an matrix, and
B, an matrix (so that
A and
B have dimensions allowing them to be multiplied in either order forming a square matrix): :\det\left(I_\mathit{m} + AB\right) = \det\left(I_\mathit{n} + BA\right), where
Im and
In are the and identity matrices, respectively. From this general result several consequences follow. {{ordered list :\det\left(I_\mathit{m} + cr\right) = 1 + rc. :\det(X + AB) = \det(X) \det\left(I_\mathit{n} + BX^{-1}A\right), : \det(X + cr) = \det(X) \det\left(1 + rX^{-1}c\right) = \det(X) + r\,\operatorname{adj}(X)\,c. }} A generalization is \det\left(Z + AWB\right) = \det\left( Z\right) \det\left(W \right) \det\left(W^{-1} + B Z^{-1} A\right)(see
Matrix determinant lemma), where
Z is an invertible matrix and
W is an invertible matrix.
Sum The determinant of the sum A+B of two square matrices of the same size is not in general expressible in terms of the determinants of
A and of
B. However, for
positive semidefinite matrices A, B and C of equal size, \det(A + B + C) + \det(C) \geq \det(A + C) + \det(B + C)\text{,} with the corollary \det(A + B) \geq \det(A) + \det(B)\text{.}
Brunn–Minkowski theorem implies that the th root of determinant is a
concave function, when restricted to
Hermitian positive-definite n\times n matrices. Therefore, if and are Hermitian positive-definite n\times n matrices, one has \sqrt[n]{\det(A+B)}\geq\sqrt[n]{\det(A)}+\sqrt[n]{\det(B)}, since the th root of the determinant is a
homogeneous function.
Sum identity for 2×2 matrices For the special case of 2\times 2 matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity: :\det(A+B) = \det(A) + \det(B) + \text{tr}(A)\text{tr}(B) - \text{tr}(AB). == Properties of the determinant in relation to other notions ==