"Linear Algebra and its Applications"-determinant Elementary transformation

To undertake the introduction of the determinant of the previous article, this article will further record about the determinant of the relevant content, including the following aspects:

(1) The proof of the Determinant 3 elementary transformations.

(2) The proof that the determinant is equal to the original determinant.

(3) theorem det (AB) = det (A) det (B) proof.

(4) The proof of Vandermonde determinant based on determinant elementary transformation.

First of all, it is worth explaining that the previous time we introduced the matrix, and did not give the Matrix line transformation of the relevant proof, in fact, according to the truth that its source is from here. Determinant and matrix are closely related, want to in this book is a matrix-based method to complete the determinant of the 3 elementary transformation of the proof.

Proof of the determinant of 3 elementary transformations:

The evidence given in the picture is a compact and indirect process, without the author's elaboration. Here the method based on Elementary matrix E is used, it is that the multiplication operation of matrix A and e just can reflect these 3 elementary transformations.

The proof that the determinant is equal to the original determinant:

This problem is very simple, but we should be able to realize the meaning of this theorem, it makes the row transformation and column transformation has the equivalence, that is, the application of the transformation of the row is applicable to the column.

A brief proving process: defining determinant A and writing out its transpose matrix a^t.

The A determinant is opened according to line I, and the A^t determinant is opened in column I, which can then be proved by the definition of transpose matrix.

Det (AB) = det (A) det (B):

On this theorem, I have previously lacked some supplementary knowledge such as "The nature of the invertible matrix", which is the derivation process | a| = | ep|...| e2| | E1| The important basis of this step transformation. The author will be taking the time to take the fundamentals of this transformation process.

Finally, the process of proving the determinant of Vandermonde.

Proving the process itself is simple, but the recursive thinking it uses is something we can learn from when solving other problems.

"Linear Algebra and its Applications"-determinant Elementary transformation