Let A be an n times n matrix Give five equivalent ways to st

Let A be an n times n matrix. Give five equivalent ways to state that the matrix A is invertible. Provide a brief justification of each of the five statements in part a.

Solution

If your matrix is a square matrix,
it must have full rank and this implies detA not equal 0.

In other case, the product of the matrices A and, in this
case A^-1 will give you a matrix of a rank equal to the
minimial rank of the matrices A and A^-1. Therefore, you
cannot obtain the identity matrix which have full rank equal n.

This implies that only classical inverse of A satisfies the equation A(A^-1)=I n.

And the rank of the matrix is not full if it has
zero column or row,
column (row) that is equal to another column (row)
column (row) is linear combination of the remaining comlumns (rows) of the matrix

I such way even if the size of the matrix is large,
we may determine if the matrix A is invertible without computation of its determinant.