How To Find Determinant Of 4x3 Matrix?


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Frances Bott answered
• Definition.
Determinants have mostly been superseded by different techniques.  Determinants are purely theoretical tools, used in checking invert ability and eigenvalues in some applications.

The determinant of a square matrix is computed using the values of the matrix itself using certain equivalent rules. 

• History.
Determinants were originally considered with no reference to the matrices. They were seen as a property instead of linear equation systems. If the system is non-zero, the determinant can 'determine' whether there is a unique solution for the system. Using this definition, the Nine Chapters on the Mathematical Arts from China, dating to around the 3rd century BC is the first use of determinants.

European mathematicians considered two-by-two determinants around the end of the 16th century by Cardano and larger ones were explored by Leibniz. Cramer worked on the theory as relative to sets of equation in 1750.  Recurrence law was added by Bezout in 1764.

IN 1771, Vandermonde first recognized determinates as sets of independent functions. Just one year later, Laplace devised generalized methods to expand determinants in terms of complementary minors.

Lagrange followed in 1773 by developing determinant theory of the second and third order, and applied determinants to elimination theory questions, proving many cases of special general identities. Further advances were made by Gauss in 1801, which not only coined the word determinants but used them extensively in the theory of numbers.

Gauss applied the term to the discriminant of a quantic, and further explored notions of reciprocal (inverse) determinants and he came close to discovering the multiplication theorem. Binet, Cauchy, Jacobi, Sylvester and others  went on to subsequently refine the theory. 

• The Leibniz/Laplaz Formula
The formula for the determinant of an n x n size matrix is the Leibniz or Laplace formula:

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