Question

Can someone explain nonsingular equivalence?

1. A is nonsingular
2. Ax=0 has only the trivial solution
3. A is row equivalent to I

The book says 1 implies 2, 2 implies 3, 3 implies 1. The only one that is intuitive to me is 3 that implies 1. Can someone explain this in plain English? Thanks

Answer 1

Only thing I remember about this is that a singular matrix has a determinant equal to zero.

Answer 2

Yes that is true too but this is presented without the use of determinants.

Answer 3

If A is non singular, then its collum vectors are linearly independent. This means that there is only one solution to the homogenius equation Ax= 0, since there is only one solution, we are row equivalent to the the nXn identity matrix, think about the shaoe if such a matrix, and its solutions to the homogenius eqaution. Every identity matrix has n linearly independent vectors, thus it is non singular.

Answer 4

shaoe if = shape of