Determinant (mathematics) | Wikipedia audio article

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00:01:03 1 Geometric meaning
00:01:55 2 Definition
00:04:02 2.1 2 × 2 matrices
00:04:23 2.2 3 × 3 matrices
00:04:55 2.3 in/i × in/i matrices
00:05:37 2.3.1 Levi-Civita symbol
00:05:58 3 Properties of the determinant
00:06:08 3.1 Schur complement
00:07:43 3.2 Multiplicativity and matrix groups
00:08:46 3.3 Laplace's formula and the adjugate matrix
00:10:32 3.4 Sylvester's determinant theorem
00:10:53 4 Properties of the determinant in relation to other notions
00:11:14 4.1 Relation to eigenvalues and trace
00:11:35 4.2 Upper and lower bounds
00:12:49 4.3 Cramer's rule
00:13:10 4.4 Block matrices
00:14:13 4.5 Derivative
00:14:34 5 Abstract algebraic aspects
00:14:45 5.1 Determinant of an endomorphism
00:15:27 5.2 Exterior algebra
00:16:40 5.2.1 Transformation on alternating multilinear in/i-forms
00:17:23 5.3 Square matrices over commutative rings and abstract properties
00:17:44 6 Generalizations and related notions
00:18:26 6.1 Infinite matrices
00:19:29 6.2 Operators in von Neumann algebras
00:20:01 6.3 Related notions for non-commutative rings
00:21:04 6.4 Further variants
00:21:14 7 Calculation
00:21:35 7.1 Decomposition methods
00:22:07 7.2 Further methods
00:22:28 8 History
00:22:49 9 Applications
00:23:21 9.1 Linear independence
00:23:52 9.2 Orientation of a basis
00:24:03 9.3 Volume and Jacobian determinant
00:24:24 9.4 Vandermonde determinant (alternant)
00:24:45 9.5 Circulants
00:25:06 10 See also
00:25:27 11 Notes
00:25:48 12 References
00:26:09 13 External links
00:26:30 History
00:26:52 Applications
00:27:02 Linear independence
00:27:44 Orientation of a basis
00:28:05 Volume and Jacobian determinant
00:28:47 Vandermonde determinant (alternant)
00:29:51 Circulants
00:31:04 See also



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SUMMARY
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In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.
In the case of a 2 × 2 matrix the determinant may be defined as









|

A

|

=


|



a


b




c


d



|


=
a
d
−
b
c
.






{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}
Similarly, for a 3 × 3 matrix A, its determinant is









|

A

|

=
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