Exercise 26

An $m\!\times\!n$ matrix has full row rank if its row rank is $m$, and it has full column rank if its column rank is $n$.

• Show that a matrix can have both full row rank and full column rank only if it is square.

• Prove that the linear system with matrix of coefficients $A$ has a solution for any $d_{1}$, …, $d_{n}$’s on the right side if and only if $A$ has full row rank.

• Prove that a homogeneous system has a unique solution if and only if its matrix of coefficients $A$ has full column rank.

• Prove that the statement “if a system with matrix of coefficients $A$ has any solution then it has a unique solution” holds if and only if $A$ has full column rank.