Question

$\large\text{what is determinant }$
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Answer 1

Answer:

Determinants are mathematical objects that are very useful in the analysis and solution of systems of linear equations. As shown by Cramer's rule, a nonhomogeneous system of linear equations has a unique solution iff the determinant of the system's matrix is nonzero (i.e., the matrix is nonsingular). For example, eliminating x, y, and z from the equations

a_1x+a_2y+a_3z = 0

(1)

b_1x+b_2y+b_3z = 0

(2)

c_1x+c_2y+c_3z = 0

(3)

gives the expression

a_1b_2c_3-a_1b_3c_2+a_2b_3c_1-a_2b_1c_3+a_3b_1c_2-a_3b_2c_1=0,  

(4)

which is called the determinant for this system of equation. Determinants are defined only for square matrices.

If the determinant of a matrix is 0, the matrix is said to be singular, and if the determinant is 1, the matrix is said to be unimodular.

The determinant of a matrix A,

|a_1 a_2 ... a_n; b_1 b_2 ... b_n; | | ... |; z_1 z_2 ... z_n|  

(5)

is commonly denoted det(A), |A|, or in component notation as sum(+/-a_1b_2c_3...), D(a_1b_2c_3...), or |a_1b_2c_3...| (Muir 1960, p. 17). Note that the notation det(A) may be more convenient when indicating the absolute value of a determinant, i.e., |det(A)| instead of ||A||. The determinant is implemented in the Wolfram Language as Det[m].

A 2×2 determinant is defined to be

det[a b; c d]=|a b; c d|=ad-bc.  

(6)

A k×k determinant can be expanded "by minors" to obtain

|a_(11) a_(12) a_(13) ... a_(1k); a_(21) a_(22) a_(23) ... a_(2k); | | | ... |; a_(k1) a_(k2) a_(k3) ... a_(kk)|=a_(11)|a_(22) a_(23) ... a_(2k); | | ... |; a_(k2) a_(k3) ... a_(kk)| -a_(12)|a_(21) a_(23) ... a_(2k); | | ... |; a_(k1) a_(k3) ... a_(kk)|+...+/-a_(1k)|a_(21) a_(22) ... a_(2(k-1)); | | ... |; a_(k1) a_(k2) ... a_(k(k-1))|.  

(7)

A general determinant for a matrix A has a value

|A|=sum_(i=1)^ka_(ij)C_(ij),  

(8)

with no implied summation over j and where C_(ij) (also denoted a^(ij)) is the cofactor of a_(ij) defined by

Answer 2

$\huge\bf\underline{\underline{\pink{A} \orange{N}\blue{S}\red{W}\green{E} \purple{R}}}$

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. ... In geometry, the signed n-dimensional volume of a n-dimensional parallelepiped is expressed by a determinant.

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