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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Determinants are used to characterize invertible matrices (namely as those matrices, and only those matrices, with non-zero determinants), and to explicitly describe the solution to a system of linear equations with Cramer's rule. They can be used to find the eigenvalues of the matrix A through the characteristic polynomial where I is the ide ...

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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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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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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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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 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 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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