The 2×2 matrix has determinant . The interpretation when the matrix has real number entries is that this gives the area of the parallelogram with vertices at (0,0), (a,c), (b,d), and (a + b, c + d), with a sign factor (which is −1 if A as a transformation matrix flips the unit square o ...

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Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

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Suppose is a square matrix. If A is a 1-by-1 matrix, then If A is a 2-by-2 matrix, then For a 3-by-3 matrix A, the formula is more complicated: For a general n-by-n matrix, the determinant was defined by Gottfried Leibniz with wha ...

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Sums and scalar products of skew-symmetric matrices are again skew-symmetric. Hence, the skew-symmetric matrices form a vector space. If matrices A and B are both skew-symmetric, then their product AB is a symmetric matrix. On the other hand, the triple product BTAB is skew-symmetric. The "skew-symmetric component" of a matrix A is the matrix B = (A − AT)/2; the "symmetric component" of A is C = (A + AT)/2; the matrix A is the ...

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The restricted Lorentz group SO+(1, 3) is isomorphic to the Möbius group, which is, in turn, isomorphic to the projective special linear group PSL(2,C). It will be convenient to work at first with SL(2,C). This group consists of all two by two complex matrices with determinant one We can write two by two Hermitian matrices in the form < ...

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Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of Lie groups. Every finite-dimensional real or complex Lie algebra arises as the Lie algebra of unique real or complex simply connected Lie group (Ado's theorem), but there may be more than one group, even more than one connected group, giving rise to the same algebra. For instance, the groups SO(3) (3×3 orthogonal matrices of determinant 1) and SU(2) (2×2 unitary matrices of determinant 1) both give rise to the sam ...

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A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, SO(n), is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, Spin(n). Likewise, O(n) has covering groups, the pin groups, Pin(n). For n > 2, Spin(n) is si ...

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Suppose we want to compute the determinant of We can go ahead and use the Leibniz formula directly: Alternatively, we can use Laplace's formula to expand the determinant along a row or column. It is best to choose a row or column with many zeros, so we will expand along the second column:

Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

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The determinant is a multiplicative map in the sense that for all n-by-n matrices A and B. This is generalized by the Cauchy-Binet formula to products of non-square matrices. It is easy to see that and thus for all n-by-n matrices A and all scalars See also:

Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

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As was pointed out above, it is possible to unambiguously define the determinant of any linear map f : V → V, if V is a finite-dimensional vector space. It makes sense to define the determinant for matrices whose entries come from any commutative ring. The computation rules, the Leibniz formula and the compatibility with matrix multiplication remain valid, except that now a matrix A is invertible if and only if See also:

Determinant, Determinant - Determinants of 2-by-2 matrices, Determinant - Applications, Determinant - General definition and computation, Determinant - Example, Determinant - Properties, Determinant - Derivative, Determinant - Generalizations and related functions, Determinant - Algorithmic implementation, Determinant - History

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