Question

Explain the similarities/differences between a coefficient matrix that is invertible and the linear independence/dependence of the columns of that matrix.

Answer 1

A linear system of equations can be of 3 types. Either the number of equations can be the same as the number of variables or less than the number of variables or, more than the number of variables.

In the first case only, there is a possibility of the coefficient matrix being invertible, as it will be a square matrix. If the coefficient matrix is invertible then the columns of this matrix will be linearly independent. The vice-versa is also true. In this case, the linear system will always have a unique solution. However, if the columns of the coefficient matrix are linearly dependent, then this matrix will not be invertible and the linear system will not have a unique solution.

In the 2^nd case, when the number of equations is less than the number of variables, the columns of the coefficient matrix cannot be linearly independent as the row rank of a matrix is equal to the column rank. Also, there will always be some free variables so that the linear system, if consistent, will have infinite solutions.

In the 3^rd case, when the number of equations is more than the number of variables, the columns of the coefficient matrix may or may not be linearly independent. In either case, the linear system, if consistent, will have infinite solutions.