Answer to Question #101038 in Linear Algebra for Frax

Question

1.Various advanced texts in linear algebra prove the following determinant criterion for rank:

The rank of a matrix A is r if and only if A has some r × r sub matrix with a nonzero determinant, and all square sub matrices of larger size have determinant zero.

(A sub matrix of A is any matrix obtained by deleting rows or columns of A. The matrix A itself is also considered to be a sub matrix of A.) Use this criterion to find the rank of the matrix.

1 0 1
A = 2 -1 4
3 -1 5

rank (A) =

2.
The rank of a matrix A is r if and only if A has some r × r sub matrix with a nonzero determinant, and all square sub matrices of larger size have determinant zero.

(A sub matrix of A is any matrix obtained by deleting rows or columns of A. The matrix A itself is also considered to be a sub matrix of A.) Use this criterion to find the rank of the matrix.

A = 1 -1 6 0
8 1 0 0
-1 6 12 0

rank (A) =

Answer 1

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