Can the basis minor of a matrix be the matrix itself?

In summary, the rank of a matrix is the maximum order of a non-zero minor. For an n x n matrix, the order of the largest non-zero minor is also n if the determinant of the whole matrix does not equal 0.
  • #1
Hello

I am trying to learn linear algebra, and I came across this definition of basis minor on this webpage:

https://en.wikibooks.org/wiki/Linear_Algebra/Linear_Dependence_of_Columns

"The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the order of the rank of the matrix is called a basis minor of the matrix, and the columns that the minor includes are called the basis columns."

Does this mean that if the determinant of the matrix does not equal zero, then its basis minor is just itself, and its rank is just the same as its order?

Thank you
 
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  • #2
That sounds right. For an ## n \times n ## matrix, the order of a minor is allowed to be the same as ## n ##. If the determinant of the whole matrix does not vanish, then rank is ## n ##, and the order of the largest non-vanishing minor is also ## n ##.
 
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  • #3
Geofleur said:
For an ## n \times n ## matrix, the order of a minor is allowed to be the same as ## n ##.

Thanks Geofleur. I thought so but just wasn't sure.
 

1. Can the basis minor of a matrix be the matrix itself?

Yes, the basis minor of a matrix can be the matrix itself. This is because the basis minor of a matrix is defined as the determinant of a square submatrix of the original matrix. If the entire matrix is selected as the submatrix, the determinant will be equal to the determinant of the original matrix, making it the basis minor.

2. What is the significance of the basis minor of a matrix?

The basis minor of a matrix is important in linear algebra as it is used to calculate important properties such as rank, invertibility, and eigenvalues. It also provides information about the linear independence of the columns or rows of the matrix.

3. How is the basis minor of a matrix calculated?

The basis minor of a matrix is calculated by selecting a square submatrix of the original matrix, finding its determinant, and then taking the absolute value of the determinant. This can be done by hand or using software such as MATLAB or Python.

4. Can the basis minor of a non-square matrix be calculated?

No, the basis minor can only be calculated for a square matrix. This is because the determinant is only defined for square matrices. However, the concept of a basis minor can be extended to rectangular matrices by considering the determinant of the submatrix with the smaller dimension.

5. How does the size of the submatrix affect the basis minor?

The size of the submatrix does not affect the basis minor, as long as it is a square submatrix. This means that the basis minor will be the same regardless of the number of rows or columns selected for the submatrix, as long as it is a square submatrix.

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