A2. Computing determinants

Answer 2, posted on April 9, 2019

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On the Web page of Wikipedia given in answer 1,

see the following part of this page:

Computing determinants:

https://en.wikipedia.org/wiki/Gaussian_elimination#Computing_determinants

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With best regards, Jean-Claude

A3. Use of elementary matrices

Answer 3, posted on April 10, 2019

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The original question is not clear:

We cannot solve an object.

We cannot solve a number.

We cannot solve the number 2.

We cannot solve a determinant.

We can solve a problem.

We can solve an equation.

We can compute a determinant.

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The following document shows that the method

of Gaussian elimination suggested by Mercedes in answer 1

can be expressed by a product of elementary matrices:

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Philippe Laval

http://ksuweb.kennesaw.edu/~plaval/

Kennesaw State University

University System of Georgia, USA

https://www.kennesaw.edu/about/

Review notes

http://ksuweb.kennesaw.edu/~plaval/review.html

Math 3260, Linear Algebra

http://ksuweb.kennesaw.edu/~plaval/review.html

Elementary matrices

http://ksuweb.kennesaw.edu/~plaval/math3260/elemmat.pdf

1.5.4 page 44, pdf-page 6:

Gaussian Elimination and Elementary Matrices

This section shows that every square matrix with complex coefficients

can be expressed as a product of elementary matrices

and a triangular matrix.

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Then we can compute the determinant of the original matrix

by using the theorem that says that

the determinant of a product of square matrices

of the same size

is equal to the product of the determinants of these matrices.

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The determinant of each of the elementary matrices in the product

is very easy to compute.

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To compute the determinant of the triangular matrix,

we use the theorem that says that

it is equal to the product of the entries along the main

diagonal of the triangular matrix.

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With best regards, Jean-Claude