EDB — 1V4

[1V4]Topics:matrix,determinant.Prerequisites:[1V2]↺↻.Difficulty:*.

We want to generalize the results of the previous exercise [1V0]↺↻ to the case of matrices \(n× n\).

Recall the following properties of the determinant of matrices \(A∈ℝ^{n× n}\).

• The rank is the dimension of the image of \(A\) (considered as a linear application from \(ℝ^ n\) to \(ℝ^ n\)) and is also the maximum number of linearly independent columns in \(A\).

• \(A\) has rank \(n\) if and only \(\det (A)≠ 0\).

• If you exchange two columns in \(A\), the determinant changes sign;

• if you add a multiple of another column to a column, the determinant does not change.

• The characterization of rank through minors, "The rank of A is equal to the highest order of an invertible minor of A".

• Laplace’s expansion of the determinant, and Jacobi’s formula (cf [1V2]↺↻).

• The determinant of \(A\) is equal to the determinant of the transpose; So every previous result holds, if you read ”row” instead of ”column”.

See also in [ 72 , 58 ] .

Show the following results.

1. Show that the gradient of the function \(\det (A)\) is not zero, if and only if the rank of \(A\) is at least \(n-1\).

2. Let \(Z\) be the set of matrices \(ℝ^{n× n}\) with null determinant. Show that it is a closed set with an empty interior.

3. Fix \(B\) a matrix with rank at most \(n-2\), show that the thesis of the theorem is false in the neighborhoods \(U_ B\) of the matrix \(B\), in the sense that \(Z∩ U_ B\) is not contained in a surface ¹ .

[[1V5]↺↻]