Question

Show that if A is a matrix with a row of zeros,then A cannot have an inverse

Answer 1

Let us show that if $A$ is a matrix with a row of zeros, then $A$ cannot have an inverse. Let $B$ be any matrix. Taking into account that $A$ is a matrix with a row of zeros, we conclude that $\det (A)=0,$ and hence $\det(AB)=\det(A)\det(B)=0\cdot\det(B)=0.$ Since for the identity matrix $E$ we have that $\det(E)=1\ne 0,$ we conclude that $AB\ne E$ for any matrix $B,$ and thus $A$ cannot have an inverse.